Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$. Dan Brânzei