HMMT 二月 2007 · 冲刺赛 · 第 23 题
HMMT February 2007 — Guts Round — Problem 23
题目详情
英文原题
- [ 12 ] In triangle ABC , ∠ ABC is obtuse. Point D lies on side AC such that ∠ ABD is right, and point Elies on side AC between A and D such that BD bisects ∠ EBC . Find CE , given that AC = 35 , BC = 7,
and BE = 5.
解析
英文解析
- [ 12 ] In triangle ABC , ∠ ABC is obtuse. Point D lies on side AC such that ∠ ABD is right, and point Elies on side AC between A and D such that BD bisects ∠ EBC . Find CE , given that AC = 35 , BC = 7,
and BE = 5.
′ ′
Answer: 10 . Reflect A and E over BD to A and E respectively. Note that the angle conditions
′ ′ ′ ′
show that A and E lie on AB and BC respectively. B is the midpoint of segment AA and CE =
′
BC − BE = 2. Menelaus’ theorem now gives
′ ′
CD AA BE
· · = 1 ,
′ ′
DA A B E Cfrom which DA = 5 CD or CD = AC/ 6. By the angle bisector theorem, DE = 5 CD/ 7, so that
CE = 12 CD/ 7 = 10.