HMMT 十一月 2013 · GEN 赛 · 第 5 题

HMMT November 2013 — GEN Round — Problem 5

[ 5 ] Let ABC be an isosceles triangle with AB = AC . Let D and E be the midpoints of segments − − → AB and AC , respectively. Suppose that there exists a point F on ray DE outside of ABC such that AB triangle BF A is similar to triangle ABC . Compute . BC

解析

[ 5 ] Let ABC be an isosceles triangle with AB = AC . Let D and E be the midpoints of segments − − → AB and AC , respectively. Suppose that there exists a point F on ray DE outside of ABC such that AB triangle BF A is similar to triangle ABC . Compute . BC √ Answer: 2 Let α = ∠ ABC = ∠ ACB , AB = 2 x , and BC = 2 y , so AD = DB = AE = EC = x ∼ and DE = y . Since 4 BF A ∼ 4 ABC and BA = AC , we in fact have 4 BF A 4 ABC , so = BF = BA = 2 x , F A = 2 y , and ∠ DAF = α . But DE ‖ BC yields ∠ ADF = ∠ ABC = α as well, √ 2 y F A AB 2 x AB x whence 4 F AD ∼ 4 ABC gives = = = = ⇒ = = 2. x AD BC 2 y BC y