HMMT 二月 2017 · 几何 · 第 5 题
HMMT February 2017 — Geometry — Problem 5
题目详情
英文原题
- Let ABCD be a quadrilateral with an inscribed circle ω and let P be the intersection of its diagonals
AC and BD . Let R , R , R , R be the circumradii of triangles AP B , BP C , CP D , DP A respectively.
1 2 3 4
If R = 31 and R = 24 and R = 12, find R .
1 2 3 4
解析
英文解析
- Let ABCD be a quadrilateral with an inscribed circle ω and let P be the intersection of its diagonals
AC and BD . Let R , R , R , R be the circumradii of triangles AP B , BP C , CP D , DP A respectively.
1 2 3 4
If R = 31 and R = 24 and R = 12, find R .
1 2 3 4
Proposed by: Sam Korsky
Answer: 19
° °
Note that ∠ AP B = 180 − ∠ BP C = ∠ CP D = 180 − ∠ DP A so sin AP B = sin BP C = sin CP D =
sin DP A . Now let ω touch sides AB, BC, CD, DA at E, F, G, H respectively. Then AB + CD =
AE + BF + CG + DH = BC + DA so
AB CD BC DA + = +
sin AP B sin CP D sin BP C sin DP Aand by the Extended Law of Sines this implies
2 R + 2 R = 2 R + 2 R
1 3 2 4
which immediately yields R = R + R − R = 19 .
4 1 3 2