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Page 1: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

MATH 3210:Euclidean and Non-Euclidean Geometry

Circular Inversion

Images of Circles and Preservation of Angles

April 15, 2020

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 2: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 3: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

O

AA′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 4: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 5: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 6: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 7: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 8: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 9: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 10: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:

(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 11: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;

(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 12: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);

(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 13: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Recall: Circular Inversion

We work in a Cartesian plane ΠF over a Euclidean ordered field (F ;<).

Γ is a fixed circle with center O and radius of length r ∈ F .

Definition. Circular inversion with respect to Γis the mapping ρΓ : ΠF \ {O} → ΠF \ {O} thatassigns to every point A 6= O the unique point A′

on ray−→OA such that OA · OA′ = r2.

Γ

OA

A′

Theorem. (1) a line containing OρΓ7→ itself,

(2) a line not containing OρΓ7→ a circle containing O,

(3) a circle containing OρΓ7→ a line not containing O.

Note: O is removed from circles and lines when we apply ρΓ.

Definition. If two circles [a line and a circle] meet ata point P, the angle between them at P is definedas the angle between their tangent lines [the lineand the tangent line of the circle] at P.

Theorem. For any circle γ, the following are equivalent:(a) γ is perpendicular to Γ;(b) ρΓ maps γ onto itself (i.e., ρΓ(γ) = γ);(c) γ contains two distinct points A,A′ such that ρΓ(A) = A′.

O

Γ

γ

A

A′

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 14: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 15: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 16: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 17: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

A

A∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A),

andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 18: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ

(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 19: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ).

Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 20: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and

• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 21: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).

• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 22: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .

• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 23: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then

�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 24: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).

� Since γ = {A∗ : A ∈ γ}, we get γ′ = δOk (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 25: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 26: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

Images of Circles under ρΓ

Theorem 1.a circle not containing O ρΓ7→ a circle not containing O.

O

Γ

γ

γ′

AA∗

A′

B = B∗

B′

Proof. Let γ be a circle not containing O,and let γ′ = ρΓ(γ).Want to show: γ′ is a circle not containing O.

For A ∈ γ let A′ = ρΓ(A), andlet A∗ be the point 6= A where line OA meets γ(let A∗ = A if OA is tangent to γ). Then

• OA · OA′ = r2 ∈ F , by the def. of ρΓ, and• OA · OA∗ = c ∈ F , independent of A, by (III.35), resp. (III.36).• Therefore, OA′ = k · OA∗ for k = r2/c ∈ F .• If O is outside γ, then�−−→OA′ =

−→OA =

−−→OA∗, so A′ = δO

k (A∗).� Since γ = {A∗ : A ∈ γ}, we get γ′ = δO

k (γ).

O

Γ

γγ′

A

A∗

A′

• If O is inside γ, the argument is similar, butsince

−−→OA′ =

−→OA and

−−→OA∗ are opposite

rays, we get γ′ = ϕ(δOk (γ)), where ϕ is the

RM that fixes O and sends every pointP 6= O to the opposite ray of

−→OP.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 27: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γ

γγ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 28: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γ

γγ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 29: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.

We starts with a lemma.

O

Γ

γγ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 30: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γ

γγ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 31: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′.

Then(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 32: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and

(2) γ0 is also tangent to γ′ at P ′.Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 33: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 34: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 35: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.

Otherwise, γ0 is a circle (say the center is C0), and� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 36: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

Page 37: MATH 3210: Euclidean and Non-Euclidean Geometryszendrei/Geom_S20/lec-04-15.pdf · Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry. Recall: Circular Inversion We

ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;

� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);

ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);

ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles: A Lemma

We will use the word ‘curve’ to mean ‘line or circle’.

Want to prove: Whenever two curves meet at an angle, their images under ρΓ meet atthe same angle.We starts with a lemma.

O

Γγ

γ′

P P′

γ0

Lemma. Let γ be a curve, γ′ = ρΓ(γ), P ∈ γ,P ′ := ρΓ(P) (∈ γ′), and assume P 6= P ′. Then

(1) there is a unique curve γ0 through P,P ′

that is tangent to γ at P, and(2) γ0 is also tangent to γ′ at P ′.

Hence, the tangent line of γ0 at P is the tangentline of γ at P, and the tangent line of γ0 at P ′ isthe tangent line of γ′ at P ′.

Proof. (1): If γ is tangent to the line OP = PP′ at P, then γ0 is this line.Otherwise, γ0 is a circle (say the center is C0), and

� γ0 is tangent to γ at P ⇔ C0 is on the line ⊥ γ through P;� P,P′ ∈ γ0 ⇔ C0 is on the perpendicular bisector of PP′.

(2): ρΓ(γ0) = γ0, because γ0 ⊥ Γ (by earlier Thm, since P,P′ ∈ γ0);ρΓ preserves tangency, because ‘tangent’ means ‘exactly one point in common’.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γ

γγ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle,

their imagesunder ρΓ meet at the same angle.

O

Γ

γγ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γ

γγ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δ

δ′

P

P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.

Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).

Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

`

`1

m

m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.

Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.

Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒

• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,

∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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ρΓ Preserves Angles

Theorem 2. Whenever two curves meet at an angle, their imagesunder ρΓ meet at the same angle.

O

Γγ

γ′

δδ′

P P′

` `1

m m1

γ0

δ0

Proof. Let γ, δ meet at P.Let P′ = ρΓ(P), γ′ = ρΓ(γ), δ′ = ρΓ(δ).Let `, m be the tangent lines to γ, δ at P.Let `1, m1 be the tangent lines to γ′, δ′ at P′.

Assume P 6= P′.Lemma⇒• there is a unique circle γ0 such that� ` is tangent to γ0 at P, and� `1 is tangent to γ0 at P′.

• there is a unique circle δ0 such that� m is tangent to δ0 at P, and� m1 is tangent to δ0 at P′.

Hence,∠(γ, δ at P) = ∠(γ0, δ0 at P) = ∠(γ0, δ0 at P′) = ∠(γ′, δ′ at P′).

For the case P = P′, see HW.

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry

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Practice Problem on Centers of CirclesWe work in a Cartesian plane ΠF over a Euclidean ordered field F .Let Γ be a fixed circle with center O and radius of length r ∈ F .

O

Γ

δδ′

Problem. Does there exist a circle δ such thatδ′ := ρΓ(δ) is a circle, and ρΓ sends the centerof δ to the center of δ′?

Circular Inversion MATH 3210: Euclidean and Non-Euclidean Geometry