HMMT 二月 2025 · 冲刺赛 · 第 6 题

HMMT February 2025 — Guts Round — Problem 6

[6] Let △ ABC be an equilateral triangle. Point D lies on segment BC such that BD = 1 and DC = 4. − → − − → Points E and F lie on rays AC and AB , respectively, such that D is the midpoint of EF . Compute EF .

解析

[6] Let △ ABC be an equilateral triangle. Point D lies on segment BC such that BD = 1 and DC = 4. − → − − → Points E and F lie on rays AC and AB , respectively, such that D is the midpoint of EF . Compute EF . Proposed by: Pitchayut Saengrungkongka √ Answer: 2 13 Solution: A E ′ C B C D F ′ ′ ′ Let C be the reflection of C over D . Then, EC ∥ C F since ECF C is a parallelogram. Thus, ′ ′ ◦ BF C is an equilateral triangle, so BF = BC = 3 and ∠ F BD = 120 . By Law of Cosines, we get √ √ √ 2 2 DF = 3 + 3 · 1 + 1 = 13 and EF = 2 13 .