HMMT 二月 2008 · 几何 · 第 9 题

英文解析

9. [ 7 ] Let ABC be a triangle, and I its incenter. Let the incircle of ABC touch side BC at D , andlet lines BI and CI meet the circle with diameter AI at points P and Q , respectively. Given BI =
   6 , CI = 5 , DI = 3, determine the value of ( DP/DQ ) .2
   Answer:75
   A64
   EFPQ
   B CI
   Let the incircle touch sides AC and AB at E and F respectively. Note that E and F both lie on d
   °
   the circle with diameter AI since ∠ AEI = ∠ AF I = 90 . The key observation is that D, E, P arecollinear. 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 .
   √ ( ) √
   2 23
   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