HMMT 二月 2023 · 几何 · 第 6 题

HMMT February 2023 — Geometry — Problem 6

Convex quadrilateral ABCD satisfies ∠ CAB = ∠ ADB = 30 , ∠ ABD = 77 , BC = CD , and ◦ ∠ BCD = n for some positive integer n . Compute n .

解析

Convex quadrilateral ABCD satisfies ∠ CAB = ∠ ADB = 30 , ∠ ABD = 77 , BC = CD , and ◦ ∠ BCD = n for some positive integer n . Compute n . Proposed by: Pitchayut Saengrungkongka Answer: 68 ◦ Solution: Let O be the circumcenter of △ ABD . From ∠ ADB = 30 , we get that △ AOB is equilateral. ◦ Moreover, since ∠ BAC = 30 , we have that AC bisects ∠ BAO , and thus must be the perpendicular bisector of BO . Therefore, we have CB = CD = CO , so C is actually the circumcenter of △ BDO . Hence, ◦ ∠ BCD = 2(180 − ∠ BOD ) ◦ = 2(180 − 2 ∠ BAD ) ◦ ◦ ◦ = 2(180 − 146 ) = 68