HMMT 十一月 2021 · THM 赛 · 第 3 题

HMMT November 2021 — THM Round — Problem 3

Let n be the answer to this problem. Hexagon ABCDEF is inscribed in a circle of radius 90. The area of ABCDEF is 8 n , AB = BC = DE = EF , and CD = F A . Find the area of triangle ABC.

解析

Let n be the answer to this problem. Hexagon ABCDEF is inscribed in a circle of radius 90. The area of ABCDEF is 8 n , AB = BC = DE = EF , and CD = F A . Find the area of triangle ABC. Proposed by: Joseph Heerens Answer: 2592 Solution: B M A C O F D E Let O be the center of the circle, and let OB intersect AC at point M ; note OB is the perpendicular bisector of AC . Since triangles ABC and DEF are congruent, ACDF has area 6 n , meaning that AOC BM 2 has area 3 n/ 2. It follows that = . Therefore OM = 54 and M B = 36, so by the Pythagorean OM 3 √ 2 2 theorem, M A = 90 − 54 = 72. Thus, ABC has area 72 · 36 = 2592.