HMMT 二月 2024 · 几何 · 第 7 题

HMMT February 2024 — Geometry — Problem 7

Let ABC be an acute triangle. Let D , E , and F be the feet of altitudes from A , B , and C to sides BC , CA , and AB , respectively, and let Q be the foot of altitude from A to line EF . Given that AQ = 20 , BC = 15 , and AD = 24 , compute the perimeter of triangle DEF .

解析

Let ABC be an acute triangle. Let D , E , and F be the feet of altitudes from A , B , and C to sides BC , CA , and AB , respectively, and let Q be the foot of altitude from A to line EF . Given that AQ = 20 , BC = 15 , and AD = 24 , compute the perimeter of triangle DEF . Proposed by: Isabella Zhu √ Answer: 8 11 Solution: A T E Q F H C B D Note that A is the excenter of △ DEF and AQ is the length of the exradius. Let T be the tangency point of the A -excircle to line DF . We have AQ = AT = 20 . It is well known that the length of DT is the semiperimeter of DEF . Note that △ ADT is a right triangle, so 2 2 2 AT + DT = AD which implies √ √ 2 2 DT = 24 − 20 = 4 11 . √ √ Thus, the perimeter of △ DEF is 2 · 4 11 = 8 11 .