HMMT 十一月 2013 · GEN 赛 · 第 5 题
HMMT November 2013 — GEN Round — Problem 5
题目详情
英文原题
- [ 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 thattriangle BF A is similar to triangle ABC . Compute .ABBC
解析
英文解析
- [ 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 thattriangle BF A is similar to triangle ABC . Compute .AB
√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 xwhence 4 F AD ∼ 4 ABC gives = = = = ⇒ = = 2.
x AD BC 2 y BC y