HMMT 二月 2009 · 几何 · 第 10 题

英文解析

10. [ 8 ] Points A and B lie on circle ω. Point P lies on the extension of segment AB past B. Line ` passesthrough P and is tangent to ω. The tangents to ω at points A and B intersect ` at points D and Crespectively. 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