MathLabs

Problem 6

Consider an isosceles triangle. Let RR be the radius of its circumcircle and rr the radius of its inscribed circle. Prove that the distance dd between the centers of these two circles is R(R−2r)\sqrt{R(R-2r)}.
Step 4 of 6: Replace BL by IL
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2Rr=AI⋅IL2Rr=AI\cdot IL
Detailed analysis

Substitute IL=BLIL=BL into the relation from similarity. This gives 2Rr=AI⋅IL2Rr=AI\cdot IL.