HMMT 二月 2014 · 几何 · 第 7 题

HMMT February 2014 — Geometry — Problem 7

Triangle ABC has sides AB = 14 , BC = 13, and CA = 15. It is inscribed in circle , which has center 0 O . Let M be the midpoint of AB , let B be the point on diametrically opposite B , and let X be the 0 intersection of AO and M B . Find the length of AX .

解析

Triangle ABC has sides AB = 14 , BC = 13, and CA = 15. It is inscribed in circle Γ, which has center ′ O . Let M be the midpoint of AB , let B be the point on Γ diametrically opposite B , and let X be the ′ intersection of AO and M B . Find the length of AX . OM BM 1 ′ ′ ◦ ′ Answer: 65/12 Since B B is a diameter, ∠ B AB = 90 , so B A ‖ OM , so = = . Thus ′ B A BA 2 ′ (13)(14)(15) AX B A 2 abc 65 = = 2, so AX = R , where R = = = is the circumradius of ABC . Putting XO OM 3 4 A 4(84) 8 65 it all together gives AX = . 12