HMMT 二月 2007 · TEAM2 赛 · 第 1 题
HMMT February 2007 — TEAM2 Round — Problem 1
题目详情
- [ 30 ] Show that P AIB is cyclic by proving that ∠ IAP is supplementary to ∠ P BI.
解析
- [ 30 ] Show that P AIB is cyclic by proving that ∠ IAP is supplementary to ∠ P BI. Solution. Note that I lies on the angle bisectors of the angles of quadrilateral ABCD. So writing ∠ DAB = 2 α, ∠ ABC = 2 β, ∠ BCD = 2 γ, and ∠ CDA = 2 δ, we have ∠ IAP + ∠ P BI = ∠ IAB + ∠ BAP + ∠ P BA + ∠ ABI = ∠ IAB + ∠ CDI + ∠ ICD + ∠ ABI = α + β + γ + δ. ◦ We are done because the angles in quadrilateral ABCD add up to 360 .