HMMT 十一月 2025 · 团队赛 · 第 6 题

HMMT November 2025 — Team Round — Problem 6

[45] Let P be a point inside triangle ABC such that BP = P C and ∠ ABP + ∠ ACP = 90 . Given that AB = 12, AC = 16, and AP = 11, compute the area of the concave quadrilateral ABP C .

解析

[45] Let P be a point inside triangle ABC such that BP = P C and ∠ ABP + ∠ ACP = 90 . Given that AB = 12, AC = 16, and AP = 11, compute the area of the concave quadrilateral ABP C . Proposed by: Pitchayut Saengrungkongka √ Answer: 96 − 10 21 Solution: © 2025 HMMT A P ′ A B C ′ ′ ′ ′ ∼ Rotate △ AP C about P to a triangle A P C , such that C coincides with B . Since △ AP C △ A P B , = ′ we find the area of ABP C is equal to the area of ABA P . The angle conditions yields ′ ′ ◦ ∠ ABA = ∠ ABP + ∠ P BA = ∠ ABP + ∠ P CA = 90 , ′ so ABA is a 12-16-20 triangle. Finally, we compute: √ 2 2 √ 12 · 16 20 · 11 − 10 ′ ′ ′ [ ABP C ] = [ ABA P ] = [ ABA ] − [ AP A ] = − = 96 − 10 21 . 2 2