HMMT 二月 2018 · 几何 · 第 5 题

HMMT February 2018 — Geometry — Problem 5

In the quadrilateral M ARE inscribed in a unit circle ω , AM is a diameter of ω , and E lies on the angle bisector of ∠ RAM . Given that triangles RAM and REM have the same area, find the area of quadrilateral M ARE .

解析

In the quadrilateral M ARE inscribed in a unit circle ω , AM is a diameter of ω , and E lies on the angle bisector of ∠ RAM . Given that triangles RAM and REM have the same area, find the area of quadrilateral M ARE . Proposed by: Yuan Yao √ 8 2 Answer: 9 Since AE bisects ∠ RAM , we have RE = EM , and E, A lie on different sides of RM . Since AM is a ◦ ◦ diameter, ∠ ARM = 90 . If the midpoint of RM is N , then from [ RAM ] = [ REM ] and ∠ ARM = 90 , we find AR = N E . Note that O , the center of ω , N , and E are collinear, and by similarity of triangles 1 1 1 2 N OM and RAM , ON = AR = N E . Therefore, ON = and N E = . By the Pythagorean 2 2 3 3 √ √ √ 4 2 1 4 2 2 8 2 theorem on triangle RAM , RM = , Therefore, the area of M ARE is 2 · · · = . 3 2 3 3 9