HMMT 十一月 2022 · 冲刺赛 · 第 25 题

HMMT November 2022 — Guts Round — Problem 25

[13] In convex quadrilateral ABCD with AB = 11 and CD = 13, there is a point P for which △ ADP and △ BCP are congruent equilateral triangles. Compute the side length of these triangles.

解析

[13] In convex quadrilateral ABCD with AB = 11 and CD = 13, there is a point P for which △ ADP and △ BCP are congruent equilateral triangles. Compute the side length of these triangles. Proposed by: Albert Wang Answer: 7 Solution: Evidently ABCD is an isosceles trapezoid with P as its circumcenter. Now, construct isosceles trape- 1 ′ ′ ′ ′ ′ ◦ zoid AB BC (that is, BB is parallel to AC .) Then AB P D is a rhombus, so ∠ B CD = ∠ B P D = 60 2 ′ ′ ◦ by the inscribed angle theorem. Also, B C = 11 because the quadrilateral B AP C is a 60 rotation √ ′ of ADP B about P . Since CD = 13, we use the law of cosines to get that B D = 7 3. Hence AP = 7.