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 2 of 5: Find the circumcenter
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In plain words

The circumcenter shares the axis.

O=(L2cos⁡α,0),R=L2cos⁡αO=\left(\frac{L}{2\cos\alpha},0\right),\quad R=\frac{L}{2\cos\alpha}
Detailed analysis

The circumcenter is on the xx-axis. Equal distances to AA and BB give O=(L/(2cos⁡α),0)O=(L/(2\cos\alpha),0) and R=L/(2cos⁡α)R=L/(2\cos\alpha).