HMMT 二月 2013 · 冲刺赛 · 第 14 题

HMMT February 2013 — Guts Round — Problem 14

[ 8 ] Consider triangle ABC with ∠ A = 2 ∠ B . The angle bisectors from A and C intersect at D , and DE 1 AB the angle bisector from C intersects AB at E . If = , compute . DC 3 AC

解析

[ 8 ] Consider triangle ABC with ∠ A = 2 ∠ B . The angle bisectors from A and C intersect at D , and DE 1 AB the angle bisector from C intersects AB at E . If = , compute . DC 3 AC 7 Answer: Let AE = x and BE = y . Using angle-bisector theorem on △ ACE we have 9 x : DE = AC : DC , so AC = 3 x . Using some angle chasing, it is simple to see that ∠ ADE = ∠ AED , 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