HMMT 二月 2017 · 几何 · 第 8 题

HMMT February 2017 — Geometry — Problem 8

Let ABC be a triangle with circumradius R = 17 and inradius r = 7. Find the maximum possible A value of sin . 2

解析

Let ABC be a triangle with circumradius R = 17 and inradius r = 7. Find the maximum possible A value of sin . 2 Proposed by: Sam Korsky √ 17+ 51 Answer: 34 Letting I and O denote the incenter and circumcenter of triangle ABC we have by the triangle in- equality that √ r AO ≤ AI + OI = ⇒ R ≤ + R ( R − 2 r ) A sin 2 and by plugging in our values for r and R we get √ A 17 + 51 sin ≤ 2 34 as desired. Equality holds when ABC is isosceles and I lies between A and O .