HMMT 十一月 2019 · 冲刺赛 · 第 25 题

HMMT November 2019 — Guts Round — Problem 25

[13] In acute 4 ABC with centroid G , AB = 22 and AC = 19. Let E and F be the feet of the altitudes ′ ′ from B and C to AC and AB respectively. Let G be the reflection of G over BC . If E , F , G , and G lie on a circle, compute BC .

解析

[13] In acute 4 ABC with centroid G , AB = 22 and AC = 19. Let E and F be the feet of the altitudes ′ ′ from B and C to AC and AB respectively. Let G be the reflection of G over BC . If E , F , G , and G lie on a circle, compute BC . Proposed by: Milan Haiman Answer: 13 ′ Note that B, C, E, F lie on a circle. Moreover, since BC bisects GG , the center of the circle that goes ′ ′ through E, F, G, G must lie on BC . Therefore, B, C, E, F, G, G lie on a circle. Specifically, the center of this circle is M , the midpoint of BC , as M E = M F because M is the center of the circumcircle BC 3 BC of BCEF . So we have GM = , which gives AM = . Then, by Apollonius’s theorem, we have 2 2 2 2 2 2 2 AB + AC = 2( AM + BM ). Thus 845 = 5 BC and BC = 13.