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 4 of 5: Locate the midpoint
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In plain words

The midpoint lies on the bisector.

AP=L1+sin⁡α,AM=Lcos⁡α1+sin⁡αAP=\frac{L}{1+\sin\alpha},\quad AM=\frac{L\cos\alpha}{1+\sin\alpha}
Detailed analysis

The tangent length is AP=scos⁡α=L/(1+sin⁡α)AP=s\cos\alpha=L/(1+\sin\alpha). Since P,QP,Q are symmetric, their midpoint MM lies on the axis and AM=APcos⁡α=Lcos⁡α/(1+sin⁡α)AM=AP\cos\alpha=L\cos\alpha/(1+\sin\alpha).