HMMT 二月 2023 · 几何 · 第 10 题

HMMT February 2023 — Geometry — Problem 10

Triangle ABC has incenter I . Let D be the foot of the perpendicular from A to side BC . Let X be a point such that segment AX is a diameter of the circumcircle of triangle ABC . Given that ID = 2, IA = 3, and IX = 4, compute the inradius of triangle ABC .

解析

Triangle ABC has incenter I . Let D be the foot of the perpendicular from A to side BC . Let X be a point such that segment AX is a diameter of the circumcircle of triangle ABC . Given that ID = 2, IA = 3, and IX = 4, compute the inradius of triangle ABC . Proposed by: Maxim Li 11 Answer: 12 Solution: Let R and r be the circumradius and inradius of ABC , let AI meet the circumcircle of ABC √ again at M , and let J be the A -excenter. We can show that △ AID ∼ △ AXJ (e.g. by bc inversion), 2 R XA IA ◦ and since M is the midpoint of IJ and ∠ AM X = 90 , IX = XJ . Thus, we have = = , so IX XJ ID 2 2 2 IX · IA 2 XI +2 AI − AX 2 2 R = = 3. But we also know R − 2 Rr = IO = . Thus, we have 2 ID 4 2 2 2 1 2 IX + 2 IA − 4 R 11 2 r = R − = . 2 R 4 12