HMMT 二月 2005 · 几何 · 第 4 题

英文解析

4. Let XY Z be a triangle with X = 60 and Y = 45 . A circle with center P passesthrough points A and B on side XY , C and D on side Y Z , and E and F on side ZX .
   Suppose AB = CD = EF . Find XP Y in degrees.6
   Solution: 255 / 21
   Since P AB , P CD , and P EF are all isosceles triangles with equal legs and equalbases, they are congruent. It follows that the heights of each are the same, so that P isequidistant from the sides of XY Z . Therefore, P is the incenter and therefore lies on
   1 ° 1
   6 6 6 6
   the angle bisectors of XY Z . Thus Y XP = Y XZ = 30 and P Y X = ZY X =
   2 2
   ° ° °
   45 45 255
   ° °
   . Therefore XP Y = 180 − 30 − = .6
   2 2 2