HMMT 二月 2013 · 冲刺赛 · 第 14 题
HMMT February 2013 — Guts Round — Problem 14
题目详情
英文原题
- [ 8 ] Consider triangle ABC with ∠ A = 2 ∠ B . The angle bisectors from A and C intersect at D , and
DE 1 ABthe angle bisector from C intersects AB at E . If = , compute .
DC 3 AC
解析
英文解析
- [ 8 ] Consider triangle ABC with ∠ A = 2 ∠ B . The angle bisectors from A and C intersect at D , and
DE 1 ABthe angle bisector from C intersects AB at E . If = , compute .
DC 3 AC
Answer: Let AE = x and BE = y . Using angle-bisector theorem on △ ACE we have 7
x : DE = AC : DC , so AC = 3 x . Using some angle chasing, it is simple to see that ∠ ADE = ∠ AED ,9
1 4
so AD = AE = x . Then, note that △ CDA ∼ △ CEB , so y : ( DC + DE ) = x : DC , so y : x = 1+ = ,
3 3
4 7 7 7
so AB = x + x = x . Thus the desired answer is AB : AC = x : 3 x = .
3 3 3 9