HMMT 二月 2017 · 几何 · 第 7 题
HMMT February 2017 — Geometry — Problem 7
题目详情
英文原题
- Let ω and Γ by circles such that ω is internally tangent to Γ at a point P . Let AB be a chord of Γtangent to ω at a point Q . Let R 6 = P be the second intersection of line P Q with Γ. If the radius of Γis 17, the radius of ω is 7, and = 3, find the circumradius of triangle AQR .AQBQ
解析
英文解析
- Let ω and Γ by circles such that ω is internally tangent to Γ at a point P . Let AB be a chord of Γtangent to ω at a point Q . Let R 6 = P be the second intersection of line P Q with Γ. If the radius of Γis 17, the radius of ω is 7, and = 3, find the circumradius of triangle AQR .AQ
Proposed by: Sam Korsky bQ
√
Answer: 170
Let r denote the circumradius of triangle AQR . By Archimedes Lemma, R is the midpoint of arc ABof Γ. Therefore ∠ RAQ = ∠ RP B = ∠ RP A so 4 RAQ ∼ 4 RP A . By looking at the similarity ratiobetween the two triangles we haver AQ
17 AP=
Now, let AP intersect ω again at X 6 = P . By homothety we have XQ ‖ AR so
AX P Q 7 10 = 1 − = 1 − =
AP P R 17 17
But we also know
AX · AP = AQ2
10 so
2 2
AP = AQ
Thus 17
√
r AQ 10 = =
17 AP 17
√
so we compute r = 170 as desired.