HMMT 二月 2009 · 几何 · 第 10 题

HMMT February 2009 — Geometry — Problem 10

[ 8 ] Points A and B lie on circle ω. Point P lies on the extension of segment AB past B. Line passes through P and is tangent to ω. The tangents to ω at points A and B intersect at points D and C respectively. Given that AB = 7 , BC = 2 , and AD = 3 , compute BP.

解析

[ 8 ] Points A and B lie on circle ω. Point P lies on the extension of segment AB past B. Line passes through P and is tangent to ω. The tangents to ω at points A and B intersect at points D and C respectively. Given that AB = 7 , BC = 2 , and AD = 3 , compute BP. Answer: 9 Solution: Say that ` be tangent to ω at point T. Observing equal tangents, write CD = CT + DT = BC + AD = 5 . Let the tangents to ω at A and B intersect each other at Q. Working from Menelaus applied to triangle CDQ and line AB gives DA QB CP − 1 = · · AQ BC P D DA CP = · BC P C + CD 3 CP = · , 2 P C + 5 2 2 from which P C = 10 . By power of a point, P T = AP · BP, or 12 = BP · ( BP + 7) , from which BP = 9 . 4