HMMT 二月 2020 · 冲刺赛 · 第 2 题

HMMT February 2020 — Guts Round — Problem 2

[4] Let ABC be a triangle and ω be its circumcircle. The point M is the midpoint of arc BC not containing A on ω and D is chosen so that DM is tangent to ω and is on the same side of AM as C . It ◦ is given that AM = AC and ∠ DM C = 38 . Find the measure of angle ∠ ACB.

解析

[4] Let ABC be a triangle and ω be its circumcircle. The point M is the midpoint of arc BC not containing A on ω and D is chosen so that DM is tangent to ω and is on the same side of AM as C . ◦ It is given that AM = AC and ∠ DM C = 38 . Find the measure of angle ∠ ACB. Proposed by: Joseph Heerens ◦ Answer: 33 ◦ ◦ ◦ Solution: By inscribed angles, we know that ∠ BAC = 38 · 2 = 76 which means that ∠ C = 104 − ∠ B . ∠ M AC ◦ ◦ Since AM = AC , we have ∠ ACM = ∠ AM C = 90 − = 71 . Once again by inscribed angles, 2 ◦ ◦ this means that ∠ B = 71 which gives ∠ C = 33 . A B C D M