HMMT 二月 2012 · TEAM2 赛 · 第 9 题

英文解析

9. [ 20 ] Let ABC be a triangle with AB < AC . Let M be the midpoint of BC . Line l is drawn through
   M so that it is perpendicular to AM , and intersects line AB at point X and line AC at point Y . Prove
   °
   that ∠ BAC = 90 if and only if quadrilateral XBY C is cyclic.
   Answer: see belowp
   Yl pA
   B C
   First, note that XBY C cyclic is equivalent to ∡ BXM = ∡ ACB . However, note that ∡ BXM =XM
   ° °
   90 − ∡ BAM , so XBY C cyclic is in turn equivalent to ∡ BAM + ∡ ACB = 90 .
   Let the line tangent to the circumcircle of △ ABC at A be p , and let P be an arbitrary point on p on
   °
   the same side of AM as B . Note that ∡ P AB = ∡ ACB . If ∡ ACB = 90 − ∡ BAM we have l ⊥ AMand thus the circumcenter O of △ ABC lies on AM . Since AB < AC , we must have O = M , and
   ° ° ° °
   ∡ BAC = 90 . Conversely, if ∡ BAC = 90 , ∡ P AM = 90 , and it follows that ∡ ACB = 90 − ∡ BAM .