HMMT 二月 2022 · 团队赛 · 第 3 题
HMMT February 2022 — Team Round — Problem 3
题目详情
- [25] Let triangle ABC be an acute triangle with circumcircle Γ. Let X and Y be the midpoints of d d minor arcs AB and AC of Γ, respectively. If line XY is tangent to the incircle of triangle ABC and the radius of Γ is R , find, with proof, the value of XY in terms of R .
解析
- [25] Let triangle ABC be an acute triangle with circumcircle Γ. Let X and Y be the midpoints of
d d
minor arcs AB and AC of Γ, respectively. If line XY is tangent to the incircle of triangle ABC and
the radius of Γ is R , find, with proof, the value of XY in terms of R .
Proposed by: Akash Das
√
Answer: 3 R
Solution: Note that X and Y are the centers of circles ( AIB ) and ( AIC ), respectively, so we have
XY perpendicularly bisects AI , where I is the incenter. Since XY is tangent to the incircle, we have
BAC ◦ d d AI has length twice the inradius. Thus, we get ∠ A = 60 . Thus, since XY = , we have XY is a 2 √ ◦ 120 arc. Thus, we have XY = R 3.