HMMT 二月 2017 · 冲刺赛 · 第 18 题

HMMT February 2017 — Guts Round — Problem 18

[ 10 ] Let ABCD be a quadrilateral with side lengths AB = 2, BC = 3, CD = 5, and DA = 4. What is the maximum possible radius of a circle inscribed in quadrilateral ABCD ?

解析

[ 10 ] Let ABCD be a quadrilateral with side lengths AB = 2, BC = 3, CD = 5, and DA = 4. What is the maximum possible radius of a circle inscribed in quadrilateral ABCD ? Proposed by: Sam Korsky √ 2 30 Answer: 7 Let the tangent lengths be a, b, c, d so that a + b = 2 b + c = 3 c + d = 5 d + a = 4 Then b = 2 − a and c = 1 + a and d = 4 − a . The radius of the inscribed circle of quadrilateral ABCD is given by √ √ 2 abc + abd + acd + bcd − 7 a + 16 a + 8 = a + b + c + d 7 √ √ 2 30 8 120 This is clearly maximized when a = which leads to a radius of = . 7 49 7