Problem 4

In a triangle

ABC

we have

AB=AC

. A circle internally tangent to the circumcircle is also tangent to

AB,AC

P,Q

. Prove that the midpoint of

PQ

is the incenter of triangle ABC.

Step 1 of 5: Set symmetric coordinates

Previous step Next step

In plain words

Symmetry leaves one coordinate.

\angle BAC=2\alpha,\quad AB=AC=L

Detailed analysis

Put $A=(0,0)$ and the angle bisector on the $x$ -axis. Then $B=(L\cos\alpha,L\sin\alpha)$ and $C=(L\cos\alpha,-L\sin\alpha)$ . The small circle has center $(s,0)$ and radius $s\sin\alpha$ .