HMMT 十一月 2023 · 冲刺赛 · 第 14 题

HMMT November 2023 — Guts Round — Problem 14

[9] Suppose that point D lies on side BC of triangle ABC such that AD bisects ∠ BAC , and let ℓ denote the line through A perpendicular to AD . If the distances from B and C to ℓ are 5 and 6, respectively, compute AD .

解析

[9] Suppose that point D lies on side BC of triangle ABC such that AD bisects ∠ BAC , and let ℓ denote the line through A perpendicular to AD . If the distances from B and C to ℓ are 5 and 6, respectively, compute AD . Proposed by: Rishabh Das 60 Answer: 11 Solution: Q A P X C B D Let ℓ , the external angle bisector, intersect BC at X . By the external angle bisector theorem, AB : AC = XB : XC = 5 : 6, so BD : DC = 5 : 6 by the angle bisector theorem. Then AD is a weighted average of the distances from B and C to ℓ , namely 6 5 60 · 5 + · 6 = . 11 11 11