HMMT 二月 2007 · 几何 · 第 10 题

HMMT February 2007 — Geometry — Problem 10

[ 8 ] ABCD is a convex quadrilateral such that AB = 2 , BC = 3 , CD = 7, and AD = 6. It also has an incircle. Given that ∠ ABC is right, determine the radius of this incircle. 1

解析

[ 8 ] ABCD is a convex quadrilateral such that AB = 2 , BC = 3 , CD = 7, and AD = 6. It also has an incircle. Given that ∠ ABC is right, determine the radius of this incircle. √ 1 + 13 2 2 2 2 2 Answer: . Note that AC = AB + BC = 13 = CD − DA . It follows that ∠ DAC is right, 3 and so √ √ [ ABCD ] = [ ABC ] + [ DAC ] = 2 · 3 / 2 + 6 · 13 / 2 = 3 + 3 13 On the other hand, if I denotes the incenter and r denotes the inradius, [ ABCD ] = [ AIB ] + [ BIC ] + [ CID ] + [ DIA ] = AB · r/ 2 + BC · r/ 2 + CD · r/ 2 + DA · r/ 2 = 9 r √ √ 1+ 13 Therefore, r = (3 + 3 13) / 9 = . 3 3