PUMaC 2014 · 几何（A 组） · 第 3 题

PUMaC 2014 — Geometry (Division A) — Problem 3

[ 4 ] Let O be the circumcenter of triangle ABC with circumradius 15. Let G be the centroid o of ABC and let M be the midpoint of BC . If BC = 18 and ∠ M OA = 150 , find the area of OM G .

解析

[ 4 ] Let O be the circumcenter of triangle ABC with circumradius 15. Let G be the centroid o of ABC and let M be the midpoint of BC . If BC = 18 and ∠ M OA = 150 , find the area of OM G . Solution: Since O is the circumcenter, we have that OM is perpendicular to BC and so OM B forms an equilateral triangle. Since OB = 15 , BM = 9 ⇒ OM + 12. Then we have that AO = 1 1 15 ⇒ [ AOM ] = (12)(15) sin 150 = 45. Then G splits AM into the ratio 2 : 1 and so 2 GM 1 [ OM G ] = [ AOM ] = 45 = 15 . AM 3