Volume 10, Number 3 August 2005 – September 2005 Famous Geometry TheoremsKin Y. Li Olympiad Corner The 2005 International MathematicalOlympiad was held in Merida, Mexico on July 13 and 14. Below are the problems. Problem 1. Six points are chosen on the sides of an equilateral triangleABC: A , A on BC; B , B on CA; C, Con AB. These points are the vertices of a convex hexagon AA BB CC with equal side lengths. Prove that the lines A B ,BCand C A are concurrent. 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Problem 2. Let a 1 , a 2 , … be a sequence of integers with infinitely many positive terms and infinitely many negative terms. Suppose that fo r e ach pos itive integern, the numbers a 1 , a 2 , …, a n leave n different remainders on division by n. Prove that each integer occurs exactly once in the sequence. Problem 3. Let x, y andzbe positive real numbers such that xyz≥ 1. Prove that . 0 2 2 5 2 5 2 2 5 2 5 2 2 5 2 5 ≥ + + − + + + − + + + − y x zzzx zy y y zy x x x (continued on page 4)Editors: (CHEUNG Pak-Hong), Munsang College, HK (KOTsz-Mei) (LEUNG Tat-Wing) (LI Kin-Yin), Dept. of Math., HKUST (NG Keng-Po Roger), ITC, HKPU Artist: (YEUNG Sau-Ying Camille), MFA, CU Acknowledgment: Thanks to Elina Chiu, Math. Dept., HKUST for general assistance. On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submis sion, please include your name, address, school, email, telephone and fax numbers (ifavailable). Electronic submissi ons, especially in MS Word , are encouraged. The deadline for receiving mater ial for the next issue is October 30, 2005. For individual subscription for the next five issues for the 03-04 academic year, send us five stamped self-addressed envelopes. Send all correspo ndence to: Dr. Kin-Yin LI Department of Mathematics The Hong Kong University of Science and Technolo gy Clear Water Bay, Kowloon, Hong Kong Fax: (852) 2358 1643 Email: [email protected]There are many famous geometry theorems. We will look at some ofthem and some of their applications. Below we will write P = WX∩ YZto denotePis the point of intersection oflines WXand YZ. If pointsA, B, Care collinear, we will introduce the sign convention : AB/ BC= BCAB / (so ifBis between A and C, then AB/BC≥ 0, otherwiseAB/BC≤ 0). Menelaus’ Theorem PointsX, Y, Zare taken from lines AB, BC, CA (which are the sides of△ ABCextended) respectively. If there is a line passing throughX, Y, Z, then . 1 − = ⋅ ⋅ ZA CZYCBYXB AXB A ZCXYProofLetL be a line perpendicular to the line throughX, Y, Zand intersect it at O. Let A’, B’, C’be the feet of the perpendiculars from A, B, Cto Lrespectively. Then . ' ' , ' ' , ' ' OA O CZA CZOCO B YCBYOB O A XB AX= = = Multiplying these equations together, we get the result. The converse of Menelaus’ Theorem is also true. T o see this, letZ’=XY∩CA. Then applying Menelaus theorem to the line throughX, Y, Z’and comparing with the equation above, we get CZ/ZA=CZ’/Z’A . It followsZ=Z’. Pascal’s Theorem LetA, B, C, D, E, Fbe points on a circle (which are not necessarily in cyclic order). Let P=AB∩ DE, Q=BC∩ EF, R=CD ∩ F A. ThenP,Q,R are collinear. YPQ B R XZEFD CA ProofLetX = EF∩ AB, Y = AB ∩ CD, Z = CD ∩ EF. Applying Menelaus’ Theorem respectively to linesBC, DE, F A cutting △ XYZextended , we have , 1 − = ⋅ ⋅ CZYCBYXB QXZQ , 1 − = ⋅ ⋅ EXZEDZYD PYXP. 1 − = ⋅ ⋅ AYXA FXZFRZYR Multiplying these three equations together, then using the intersecting chord theorem (see vol4, no. 3, p. 2 ofMathematical Excalibur) to getXA·XB = XE·XF, YC·YD = YA·YB, ZE·ZF = ZC·ZD, we arrive at the equation . 1 − = ⋅ ⋅ RZYR PYXPQXZQ By the converse of Menelaus’ Theorem, this implies P, Q, R are collinear. We remark that there are limiting cases of Pascal’ s Theorem. For example, we may move A to approach B. In the limit, A andB will coincide and the line AB will become the tangent line at B. Below we will give some examples ofusing Pascal’s Theorem in geometry problems. Example 1 (2001 Macedonian Math Olympiad) For the circumcircle of△ ABC, let D be the intersection of the tangent line atA with lineBC, Ebe the intersection of the tangent line at B with line CA andFbe the intersection of the tangent line at Cwith line AB. Prove that points D, E, Fare collinear.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.