HMMT 二月 2014 · 几何 · 第 8 题

HMMT February 2014 — Geometry — Problem 8

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 Y is tangent to AN . Find the area of triangle N AY .

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 Y is 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 √ √ 2 2 (10 / 3) 1 1 73 GA 100 2 2 2 √ √ CM = 8 + 3 = . By power of a point, ( GM )( GY ) = GA , so GY = = = . 73 3 3 3 GY 3 73 3 Thus [ GAY ] [ N AY ] [ N AY ] = [ GAM ] · · [ GAM ] [ GAY ] 1 GY N A = [ ABC ] · · 6 GM GA 100 3 600 = 4 · · = 73 2 73