HMMT 二月 2017 · 几何 · 第 7 题

HMMT February 2017 — Geometry — Problem 7

Let ω and Γ by circles such that ω is internally tangent to Γ at a point P . Let AB be a chord of Γ tangent to ω at a point Q . Let R 6 = P be the second intersection of line P Q with Γ. If the radius of Γ AQ is 17, the radius of ω is 7, and = 3, find the circumradius of triangle AQR . BQ

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 Γ tangent to ω at a point Q . Let R 6 = P be the second intersection of line P Q with Γ. If the radius of Γ AQ is 17, the radius of ω is 7, and = 3, find the circumradius of triangle AQR . BQ Proposed by: Sam Korsky √ Answer: 170 Let r denote the circumradius of triangle AQR . By Archimedes Lemma, R is the midpoint of arc AB of Γ. Therefore ∠ RAQ = ∠ RP B = ∠ RP A so 4 RAQ ∼ 4 RP A . By looking at the similarity ratio between the two triangles we have r 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 2 AX · AP = AQ so 10 2 2 AP = AQ 17 Thus √ r AQ 10 = = 17 AP 17 √ so we compute r = 170 as desired.