HMMT 十一月 2020 · 冲刺赛 · 第 27 题

HMMT November 2020 — Guts Round — Problem 27

[13] In 4 ABC , D and E are the midpoints of BC and CA , respectively. AD and BE intersect at G . Given that GECD is cyclic, AB = 41, and AC = 31, compute BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMO 2020, November 14–21, 2020 — GUTS ROUND Organization Team Team ID#

解析

[13] In 4 ABC , D and E are the midpoints of BC and CA , respectively. AD and BE intersect at G . Given that GECD is cyclic, AB = 41, and AC = 31, compute BC . Proposed by: Sheldon Kieren Tan Answer: 49 Solution: A E G B C D By Power of a Point, 2 2 2 1 AD = AD · AG = AE · AC = · 31 3 2 2 3 2 so AD = · 31 . The median length formula yields 4 2 2 2 2 1 AD = (2 AB + 2 AC − BC ) , 4 whence √ √ 2 2 2 2 2 2 BC = 2 AB + 2 AC − 4 AD = 2 · 41 + 2 · 31 − 3 · 31 = 49 .