Power Theorem
Given a pointPand a circle, pass two lines throughPthat intersect the circle in pointsAandDand, respectively,BandC.ThenAPDP=BPCP.The pointPmay lie either inside or outside the circle. The line throughAandD(or that throughBandCor both) may be tangent to the circle, in which caseAandDcoalesce into a single point. In all the cases, the theorem holds and is known as thePower of a Point Theorem.When the pointPis inside the circle, the theorem is also known as theTheorem of Intersecting Chords(or theIntersecting Chords Theorem) and has abeautiful interpretation. When the pointPis outside the circle, the theorem becomes theTheorem of Intersecting Secants(or theIntersecting Secants Theorem.)The proof is exactly the same in all three cases mentioned above. Since triangles ABP and CDP are similar, the following equality holds:APCP=BPDP,which is equivalent to the statement of the theorem:APDP=BPCP.The common value of the products then depends only onPand the circle and is known as thePower of PointPwith respect to the (given) circle. Note that, whenPlies outside the circle, its power equals the length of the square of the tangent from P to the circle. For example, ifB=Cso thatBPis tangent to the circleAPDP=BP2.Sometimes it is useful to employsigned segments. The convenience is that it is possible to tell points inside the circle from the points outside the circle. The power of a point inside the circle is negative, whereas that of a point outside the circle is positive. This is exactly what one obtains from thealgebraic definitionof the power of a point.The theorem is reversible: Assume pointsA,B,C,andDare not collinear. LetPbe the intersection ofADandBCsuch thatAPDP=BPCP.Then the four pointsA,B,C,andDare concyclic. To see that draw a circle through, say,A,B,andC.Assume it intersectsAPatD.Then, as was shown above,APDP=BPCP,from whichD=D.(If, say,BandCcoincide, draw the circle throughAtangent toPBatB.)
Power of a Point Theorem:
Given circle O, pointPnot on the circle, and a line throughPintersecting the circle in two points. The product of the length fromPto the first point of intersection and the length fromPto the second point of intersection is constant for any choice of a line throughPthat intersects the circle. Thisconstantis called the "power of pointP".
IfPis outside the circle ....
This becomes the theorem we know as the theorem of intersecting secants.IfPis inside the circle ....
This becomes the theorem we know as the theorem of intersecting chords.
Special Cases:Should one of the lines betangentto the circle, point A will coincide with point D, and the theorem still applies.
This becomes the theorem we know as the theorem of secant-tangent theorem.Should both of the lines be tangents to the circle, point A coincides with point D, point C coincides with point B, and the theorem still applies.
This becomes the theorem we know as the theorem of two tangents.
1.1. Given
2.2. Two points determine exactly one line.
3.3. Reflexive Property (Identity)
4.4. In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.
5.5. Substitution (or Transitive)
6.6. Congruent angles are angles of equal measure.
7.7. AA (If two angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar.
8.8. Corresponding sides of similar triangles are in proportion.
9.9. In a proportion, the product of the means equals the product of the extremes.
Theorem:For all,
Proof:
You can use the Secant-Secant Power Theorem to solve some circle problems. This theorem involves are you sitting down two secants! (If youre trying to come up with a creative name for your child like Dweezil or Moon Unit, talk to Frank Zappa, not the guy who named the power theorems.)Secant-Secant Power Theorem:If two secants are drawn from an external point to a circle, then the product of the measures of one secants external part and that entire secant is equal to the product of the measures of the other secants external part and that entire secant. (Whew!)
For instance, in the above figure,4(4 + 2) = 3(3 + 5)The following problem uses two power theorems:
Given: Diagram as shownSegmentBAis tangent to circleHatAFind:xandyThe figure includes a tangent and some secants, so look to your Tangent-Secant and Secant-Secant Power Theorems.
Now use the Secant-Secant Power Theorem with secants segmentECand segmentEGto solve fory:
A segment cant have a negative length, soy= 3. That does it.
POWER THEOREM
Theorem:For all,
Proof:As mentioned in 5.8, physicalpowerisenergy per unit time.7.19For example, when aforceproduces a motion, the power delivered is given by theforcetimes thevelocityof the motion. Therefore, ifandare in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their productis proportional to thepower per sampleat time, andbecomes proportional to the totalenergysupplied (or absorbed) by the driving force. By the power theorem,can be interpreted as theenergy per binin theDFT, orspectral power,i.e., the energy associated with a spectralbandof width.7.20\Circle PowerThepowerof a fixed pointwith respect to acircleofradiusand centeris defined by the product(1)
whereandare the intersections of a line throughwith the circle. The term "power" was first used in this way by Jacob Steiner (Steiner 1826; Coxeter and Greitzer 1967, p.30). Amazingly,(sometimes written) isindependentof the choice of the line(Coxeter 1969, p.81).
Now consider a pointnot necessarily on the circumference of the circle. Ifis the distance betweenand the circle's center, then the power of the pointrelative to the circle is(2)
Ifis outside thecircle, its power ispositiveand equal to the square of the length of the segmentfromto the tangentto thecirclethrough,(3)
Iflies along thex-axis, then the anglearound the circle at whichlies is given by solving(4)
for, giving(5)
for coordinates(6)
The pointsandareinverse points, also called polar reciprocals, with respect to theinversion circleif(7)
(Wenninger 1983, p.2).Ifis inside thecircle, then the power isnegativeand equal to the product of thediametersthrough.The powers of circle of radiuswith center having trilinear coordinateswith respect to the vertices of areference triangleare(8)
(9)
(10)
(P.Moses, pers. comm., Jan.26, 2005). Thecircle functionof such a circle is then given by(11)
Thelocusof points havingpowerwith regard to a fixedcircleofradiusis aconcentriccircleofradius. Thechordal theoremstates that thelocusof points having equalpowerwith respect to two given nonconcentriccirclesis a line called theradical line
We already know that in a circle the measure of a central angle is equal to the measure of the arc it intercepts. But what if the central angle had its vertex elsewhere?An angle whose vertex lies on a circle and whose sides intercept the circle (the sides contain chords of the circle) is called an inscribed angle. The measure of an inscribed angle is half the measure of the arc it intercepts.
Figure %: The inscribed angle measures half of the arc it interceptsIf the vertex of an angle is on a circle, but one of the sides of the angle is contained in a line tangent to the circle, the angle is no longer an inscribed angle. The measure of such an angle, however, is equal to the measure of an inscribed angle. It is equal to one-half the measure of the arc it intercepts.
Figure %: An angle whose sides are a chord and a tangent segmentThe angle ABC is equal to half the measure of arc AB (the minor arc defined by points A and B, of course).An angle whose vertex lies in the interior of a circle, but not at its center, has rays, or sides, that can be extended to form two secant lines. These secant lines intersect each other at the vertex of the angle. The measure of such an angle is half the sum of the measures of the arcs it intercepts.
Figure %: An angle whose vertex is in the interior of a circleThe measure of angle 1 is equal to half the sum of the measures of arcs AB and DE.When an angle's vertex lies outside of a circle, and its sides don't intersect with the circle, we don't necessarily know anything about the angle. The angle's sides, however, can intersect with the circle in three different ways. Its sides can be contained in two secant lines, one secant line and one tangent line, or two tangent lines. In any case, the measure of the angle is one-half the difference between the measures of the arcs it intercepts. Each case is pictured below.
Figure %: An angle whose vertex lies outside of a circleIn part (A) of the figure above, the measure of angle 1 is equal to one-half the difference between the measures of arcs JK and LM. In part (B), the measure of angle 2 is equal to one-half the difference between the measures of arcs QR and SR. In part (C), the measure of angle 3 is equal to one-half the difference between the measures of arcs BH and BJH. In this case, J is a point labeled just to make it easier to understand that when an angle's sides are parts of lines tangent to a circle, the arcs they intercept are the major and minor arc defined by the points of tangency. Here, arc BJH is the major arc.
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