HMMT 二月 2008 · 几何 · 第 9 题

HMMT February 2008 — Geometry — Problem 9

[ 7 ] Let ABC be a triangle, and I its incenter. Let the incircle of ABC touch side BC at D , and let lines BI and CI meet the circle with diameter AI at points P and Q , respectively. Given BI = 2 6 , CI = 5 , DI = 3, determine the value of ( DP/DQ ) .

解析

[ 7 ] Let ABC be a triangle, and I its incenter. Let the incircle of ABC touch side BC at D , and let lines BI and CI meet the circle with diameter AI at points P and Q , respectively. Given BI = 2 6 , CI = 5 , DI = 3, determine the value of ( DP/DQ ) . 75 Answer: 64 A Q P F E I B C D Let the incircle touch sides AC and AB at E and F respectively. Note that E and F both lie on ◦ the circle with diameter AI since ∠ AEI = ∠ AF I = 90 . The key observation is that D, E, P are collinear. To prove this, suppose that P lies outside the triangle (the other case is analogous), then 1 ◦ 1 ∠ P EA = ∠ P IA = ∠ IBA + ∠ IAB = ( ∠ B + ∠ A ) = 90 − ∠ C = ∠ DEC , which implies that D, E, P 2 2 are collinear. Similarly D, F, Q are collinear. Then, by Power of a Point, DE · DP = DF · DQ . So DP/DQ = DF/DE . √ ( ) √ 3 2 2 Now we compute DF/DE . Note that DF = 2 DB sin ∠ DBI = 2 6 − 3 = 3 3, and DE = 6 √ √ ( ) 3 24 5 3 2 2 2 DC sin ∠ DCI = 2 5 − 3 = . Therefore, DF/DE = . 5 5 8