MathLabs

Problem 4

In a triangle ABCABC we have AB=ACAB=AC. A circle internally tangent to the circumcircle is also tangent to AB,ACAB,AC at P,QP,Q. Prove that the midpoint of PQPQ is the incenter of triangle ABC.
Step 5 of 5: Identify the incenter
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In plain words

Same axis and distance identify M with the incenter.

AI=Lcos⁡α1+sin⁡α=AMAI=\frac{L\cos\alpha}{1+\sin\alpha}=AM
Detailed analysis

The area is L2sin⁡αcos⁡αL^2\sin\alpha\cos\alpha and semiperimeter L(1+sin⁡α)L(1+\sin\alpha), so r=Lsin⁡αcos⁡α/(1+sin⁡α)r=L\sin\alpha\cos\alpha/(1+\sin\alpha) and AI=r/sin⁡α=Lcos⁡α/(1+sin⁡α)=AMAI=r/\sin\alpha=L\cos\alpha/(1+\sin\alpha)=AM. Thus MM is the incenter.