Problem 6

Consider an isosceles triangle. Let

R

be the radius of its circumcircle and

r

the radius of its inscribed circle. Prove that the distance

d

between the centers of these two circles is

\sqrt{R(R-2r)}

Step 1 of 6: Set up the auxiliary points

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O=\text{circumcenter},\quad I=\text{incenter},\quad AI\cap\text{(circumcircle)}=L,\quad LO\cap\text{(circumcircle)}=M

Detailed analysis

Although the problem assumes an isosceles triangle, work with an arbitrary triangle $ABC$ . Let $O$ and $I$ be its circumcenter and incenter. Extend $AI$ to meet the circumcircle again at $L$ , then extend $LO$ to meet the circumcircle again at $M$ .