HMMT 二月 2023 · 几何 · 第 8 题

HMMT February 2023 — Geometry — Problem 8

Triangle ABC with ∠ BAC > 90 has AB = 5 and AC = 7. Points D and E lie on segment BC such ◦ that BD = DE = EC . If ∠ BAC + ∠ DAE = 180 , compute BC .

解析

Triangle ABC with ∠ BAC > 90 has AB = 5 and AC = 7. Points D and E lie on segment BC such ◦ that BD = DE = EC . If ∠ BAC + ∠ DAE = 180 , compute BC . Proposed by: Maxim Li √ Answer: 111 1 Solution: Let M be the midpoint of BC , and consider dilating about M with ratio − . This takes 3 ′ ′ B to E , C to D , and A to some point A on AM with AM = 3 A M . Then the angle condition implies ′ ◦ ′ ∠ DAE + ∠ EA D = 180 , so ADA E is cyclic. Then by power of a point, we get 2 2 AM BC ′ = AM · A M = DM · EM = . 3 36 2 2 2 2 2 2 2 2 AB +2 AC − BC 2 AB +2 AC − BC BC 2 But we also know AM = , so we have = , which rearranges to 4 12 36 √ 3 2 2 2 BC = ( AB + AC ). Plugging in the values for AB and AC gives BC = 111. 2