HMMT 二月 2016 · 几何 · 第 2 题

HMMT February 2016 — Geometry — Problem 2

Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let H be the orthocenter of ABC . Find the distance between the circumcenters of triangles AHB and AHC .

解析

Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let H be the orthocenter of ABC . Find the 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 C tions 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 B tively. 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 length 1 of the midline of triangle ABC that is parallel to BC , meaning the distance is 2( BC ) = 14. 2