PUMaC 2014 · 个人决赛（B 组） · 第 1 题

PUMaC 2014 — Individual Finals (Division B) — Problem 1

Let A, B be two points on circle γ . At point A and B we construct tangents to γ , AC and BD respectively such that the tangents are both in the clockwise direction. Let the intersection between AB and CD be P . If AC = BD , prove that P bisects the line CD .

解析

Let A, B be two points on circle . At point A and B we construct tangents to , AC and BD respectively such that the tangents are both in the clockwise direction. Let the intersection between AB and CD be P . If AC = BD , prove that P bisects the line CD . Solution: Extend DB to Q as above such that QB = BD = AC . Since AC and BQ are equal tangents, by symmetry we see that AB//QC and therefore BP//QC . Hence 4 QDC ' 4 BDP and CP QB thus = = 1 and we are done. P D BD