HMMT 二月 2025 · 几何 · 第 1 题

HMMT February 2025 — Geometry — Problem 1

Equilateral triangles △ ABC and △ DEF are drawn such that points B , E , F , and C lie on a line in this order, and point D lies inside triangle △ ABC . If BE = 14, EF = 15, and F C = 16, compute AD .

解析

Equilateral triangles △ ABC and △ DEF are drawn such that points B , E , F , and C lie on a line in this order, and point D lies inside triangle △ ABC . If BE = 14, EF = 15, and F C = 16, compute AD . Proposed by: Jackson Dryg, Karthik Venkata Vedula Answer: 26 Solution: A X D B C E F Extend DE to meet AC at X . Observe that ABEX and DF CX are isosceles trapezoids (both with ◦ base angles of 60 ), so we have • AX = BE = 14, • DX = F C = 16, and ◦ • ∠ AXD = 120 . By Law of Cosines on △ ADX , the answer is p 2 2 ◦ AD = AX + DX − 2 cos(120 ) AX · DX p 2 2 = 14 + 16 + 14 · 16 = 26 .