HMMT 十一月 2012 · GEN 赛 · 第 9 题

HMMT November 2012 — GEN Round — Problem 9

[ 7 ] Consider triangle ABC where BC = 7, CA = 8, and AB = 9. D and E are the midpoints of BC ′ ′ and CA , respectively, and AD and BE meet at G . The reflection of G across D is G , and G E meets CG at P . Find the length P G .

解析

[ 7 ] Consider triangle ABC where BC = 7, CA = 8, and AB = 9. D and E are the midpoints of BC ′ ′ and CA , respectively, and AD and BE meet at G . The reflection of G across D is G , and G E meets CG at P . Find the length P G . √ 145 1 ′ ′ Answer: Observe that since G is a reflection and GD = AG , we have AG = GG and 9 2 ′ 1 therefore, P is the centroid of triangle ACG . Thus, extending CG to hit AB at F , P G = CG = 3 √ √ 2 2 2 2(8 +7 ) − 9 2 2 145 CF = = by the formula for the length of a median. 9 9 4 9