HMMT 二月 2005 · 几何 · 第 8 题

HMMT February 2005 — Geometry — Problem 8

Let T be a triangle with side lengths 26, 51, and 73. Let S be the set of points inside T which do not lie within a distance of 5 of any side of T . Find the area of S .

解析

Let T be a triangle with side lengths 26, 51, and 73. Let S be the set of points inside T which do not lie within a distance of 5 of any side of T . Find the area of S . Solution: 135/28 Note that the sides of S are parallel to the sides of T , so S is a triangle similar to T . 1 The semiperimeter of T is s = (26 + 51 + 73) = 75. By Heron’s formula, the area 2 √ of T is 75 · 49 · 24 · 2 = 420. If r is the inradius of T , then the area of T is rs , so r = 420 / 75 = 28 / 5. It follows that the inradius of S is r − 5 = 3 / 5, and the ratio of 2 similitude between S and T is 3 / 28. Therefore, the area of S is 420 · (3 / 28) = 135 / 28.