HMMT 二月 2007 · TEAM2 赛 · 第 2 题
HMMT February 2007 — TEAM2 Round — Problem 2
题目详情
英文原题
- [ 20 ] The four sides of quadrilateral ABCD are equal in length. Determine the perimeter of ABCDgiven that it has area 120 and AC = 10 .
解析
英文解析
- [ 40 ] Show that triangle P AI is similar to triangle BIC. Then conclude that
P I
P A = · BI.
Solution. We have ∠ IBC = ∠ ABI because I lies on the angle bisector, and ∠ ABI = ∠ AP I because bC
P AIB is cyclic. Additionally,
∠ BCI = ∠ ICD = ∠ P BA = ∠ P IA,
by the angle bisector CI , that triangles P AB and IDC are similar, and the fact that P AIB iscyclic, respectively. It follows that triangles P AI and BIC are similar. In particular, it follows that
IP/P A = BC/BI , as required. 3