HMMT 二月 2020 · 团队赛 · 第 3 题

HMMT February 2020 — Team Round — Problem 3

[25] Let ABC be a triangle inscribed in a circle ω and ` be the tangent to ω at A . The line through B parallel to AC meets ` at P , and the line through C parallel to AB meets ` at Q . The circumcircles of ABP and ACQ meet at S 6 = A . Show that AS bisects BC .

解析

[25] Let ABC be a triangle inscribed in a circle ω and ` be the tangent to ω at A . The line through B parallel to AC meets ` at P , and the line through C parallel to AB meets ` at Q . The circumcircles of ABP and ACQ meet at S 6 = A . Show that AS bisects BC . Proposed by: Andrew Gu Solution 1: In directed angles, we have ] CBP = ] BCA = ] BAP, so BC is tangent to the circumcircle of ABP . Likewise, BC is tangent to the circumcircle of ACQ . Let 2 2 M be the midpoint of BC . Then M has equal power M B = M C with respect to the circumcircles of ABP and ACQ , so the radical axis AS passes through M . Q A P S B C M ′ A Solution 2: Since ] CBP = ] BCA = ] BAP = ] CQP, ′ ′ quadrilateral BCQP is cyclic. Then AS , BP , and CQ concur at a point A . Since A B ‖ AC and ′ ′ ′ A C ‖ AB , quadrilateral ABA C is a parallelogram so line ASA bisects BC .