HMMT 十一月 2016 · THM 赛 · 第 3 题

英文解析

3. The three points A , B , C form a triangle. AB = 4, BC = 5 AC = 6. Let the angle bisector of ∠ Aintersect side BC at D . Let the foot of the perpendicular from B to the angle bisector of ∠ A be E .
   Let the line through E parallel to AC meet BC at F . Compute DF .
   Proposed by: Allen Liu
   Answer: 1 / 2
   AB AC
   Since AD bisects ∠ A , by the angle bisector theorem = , so BD = 2 and CD = 3. Extend BE
   BD CDto hit AC at X . Since AE is the perpendicular bisector of BX , AX = 4. Since B, E, X are collinear,
   applying Menelaus’ Theorem to the triangle ADC , we have
   AE DB CX
   · · = 1
   ED BC XA
   AE DF DE DC 1
   This implies that = 5, and since EF ‖ AC , = , so DF = = .
   ED DC DA 6 2