HMMT 二月 2016 · 几何 · 第 2 题

英文解析

2. Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let H be the orthocenter of ABC . Findthe distance between the circumcenters of triangles AHB and AHC .
   Proposed by: Evan Chen
   Answer: 14
   Let H be the reflection of H over AC and let H be the reflection of H over AB . The reflec b Ctions of H over AB, AC lie on the circumcircle of triangle ABC . Since the circumcenters of triangles
   AH B, AH C are both O , the circumcenters of AHB, AHC are reflections of O over AB, AC respec c Btively. Moreover, the lines from O to the circumcenters in question are the perpendicular bisectors of
   AB and AC . Now we see that the distance between the two circumcenters is simply twice the lengthof the midline of triangle ABC that is parallel to BC , meaning the distance is 2( BC ) = 14.1 2