HMMT 十一月 2012 · 团队赛 · 第 8 题

英文解析

8. [ 4 ] ABC is a triangle with AB = 15, BC = 14, and CA = 13. The altitude from A to BC is extendedto meet the circumcircle of ABC at D . Find AD .
   Answer: Let the altitude from A to BC meet BC at E . The altitude AE has length 12; one 63
   way to see this is that it splits the triangle ABC into a 9 − 12 − 15 right triangle and a 5 − 12 − 134
   right triangle; from this, we also know that BE = 9 and CE = 5.
   Now, by Power of a Point, AE · DE = BE · CE , so DE = ( BE · CE ) /AE = (9 · 5) / (12) = 15 / 4. Itthen follows that AD = AE + DE = 63 / 4.