HMMT 二月 2014 · 几何 · 第 7 题
HMMT February 2014 — Geometry — Problem 7
题目详情
英文原题
- Triangle ABC has sides AB = 14 , BC = 13, and CA = 15. It is inscribed in circle , which has center
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 .0
解析
英文解析
- 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
it all together gives AX = .65
12