HMMT 二月 2007 · TEAM2 赛 · 第 1 题

英文解析

1. [ 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 .