HMMT 二月 2019 · 冲刺赛 · 第 14 题

HMMT February 2019 — Guts Round — Problem 14

[ 7 ] Let ABC be a triangle where AB = 9 , BC = 10 , CA = 17. Let Ω be its circumcircle, and let A , B , C be the diametrically opposite points from A, B, C, respectively, on Ω . Find the area of the 1 1 1 convex hexagon with the vertices A, B, C, A , B , C . 1 1 1

解析

[ 7 ] Let ABC be a triangle where AB = 9 , BC = 10 , CA = 17. Let Ω be its circumcircle, and let A , B , C be the diametrically opposite points from A, B, C, respectively, on Ω . Find the area of the 1 1 1 convex hexagon with the vertices A, B, C, A , B , C . 1 1 1 Proposed by: Yuan Yao 1155 Answer: 4 2 2 2 9 − 17 − 10 15 8 We first compute the circumradius of ABC : Since cos A = = − , we have sin A = and 2 · 9 · 17 17 17 a 170 1 R = = . Moreover, we get that the area of triangle ABC is bc sin A = 36. 2 sin A 16 2 Note that triangle ABC is obtuse, The area of the hexagon is equal to twice the area of triangle ABC (which is really [ ABC ] + [ A B C ]) plus the area of rectangle ACA C . The dimensions of ACA C 1 1 1 1 1 1 1 √ 51 51 1155 2 2 are AC = 17 and A C = (2 R ) − AC = , so the area of the hexagon is 36 · 2 + 17 · = . 1 4 4 4