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 2 of 6: Relate a radius of the incircle to the auxiliary chord

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D\in AB\cap\text{(incircle)},\quad\angle ADI=\angle MBL=90^\circ,\quad\angle IAD=\angle LMB

Detailed analysis

Let $D$ be the tangency point of the incircle with $AB$ . The right angles and the equal angles at $A$ and $M$ give $\triangle ADI\sim\triangle MBL$ . Since $ID=r$ and $ML=2R$ , similarity yields $ID\cdot ML=AI\cdot BL$ , hence $2Rr=AI\cdot BL$ .