HMMT 二月 2014 · 几何 · 第 8 题
HMMT February 2014 — Geometry — Problem 8
题目详情
英文原题
- Let ABC be a triangle with sides AB = 6, BC = 10, and CA = 8. Let M and N be the midpoints of
BA and BC , respectively. Choose the point Y on ray CM so that the circumcircle of triangle AM Yis tangent to AN . Find the area of triangle N AY .
解析
英文解析
- Let ABC be a triangle with sides AB = 6, BC = 10, and CA = 8. Let M and N be the midpoints of
BA and BC , respectively. Choose the point Y on ray CM so that the circumcircle of triangle AM Yis tangent to AN . Find the area of triangle N AY .
2 10
Answer: 600/73 Let G = AN ∩ CM be the centroid of ABC . Then GA = GN = and GM =
3 3
√
√
(10 / 3)22
1 1 73 GA 100
2 22
√
√
CM = 8 + 3 = . By power of a point, ( GM )( GY ) = GA , so GY = = = .
3 3 3 GY73
3 73
Thus 3
[ GAY ] [ N AY ]
[ N AY ] = [ GAM ] · ·
[ GAM ] [ GAY ]
1 GY N A = [ ABC ] · ·
6 GM GA
100 3 600 = 4 · · =
73 2 73