HMMT 二月 2024 · 冲刺赛 · 第 6 题

HMMT February 2024 — Guts Round — Problem 6

[6] In triangle ABC , points M and N are the midpoints of AB and AC , respectively, and points P and Q trisect BC . Given that A , M , N , P , and Q lie on a circle and BC = 1 , compute the area of triangle ABC . b c a

解析

[6] In triangle ABC , points M and N are the midpoints of AB and AC , respectively, and points P and Q trisect BC . Given that A , M , N , P , and Q lie on a circle and BC = 1 , compute the area of triangle ABC . Proposed by: Rishabh Das √ 7 Answer: 12 Solution: Note that M P ∥ AQ , so AM P Q is an isosceles trapezoid. In particular, we have AM = 1 2 2 M B = BP = P Q = , so AB = . Thus ABC is isoceles with base 1 and legs , and the height from 3 3 3 √ √ 7 7 A to BC is , so the area is . 6 12 A M N B C Q P b c a