HMMT 二月 2017 · 几何 · 第 8 题

英文解析

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