Solutions to Problems 1191–1200

Q1191 Let $ABC$ be a triangle with sides $a,b,c$ in the usual way and let $r$ be its circumradius.

  1. Show that $\displaystyle 3r > \frac{a+b+c}{2}.$
  2. Let $P$ be any point within $ABC$, and let $r_1$ be the circumradius of $ABP.$  Is it true that $r_1 < r?$