HMMT 二月 2011 · TEAM2 赛 · 第 9 题
HMMT February 2011 — TEAM2 Round — Problem 9
题目详情
英文原题
- [ 25 ] Let ABC be a non-isosceles, non-right triangle, let ω be its circumcircle, and let O be its circumcenter. Let M be the midpoint of segment BC . Let the tangents to ω at B and C intersect at X .
Prove that ∠ OAM = ∠ OXA . (Hint: use SAS similarity).
解析
英文解析
- [ 25 ] Let ABC be a non-isosceles, non-right triangle, let ω be its circumcircle, and let O be its circumcenter. Let M be the midpoint of segment BC . Let the tangents to ω at B and C intersect at X .
Prove that ∠ OAM = ∠ OXA . (Hint: use SAS similarity).
Team Round B
MHOA
B C
Solution:XDπ OM OC
Note that 4 OM C ∼ 4 OCX since ∠ OM C = ∠ OCX = . Hence = , or, equivalently,
2 OC OX
OM OA = . By SAS similarity, it follows that 4 OAM ∼ 4 OXA . Therefore, ∠ OAM = ∠ OXA .
OA OX