Problem 2

Prove that if

n\ge4

, every quadrilateral that can be inscribed in a circle can be dissected into

n

quadrilaterals each of which is inscribable in a circle.

Step 3 of 4: Complete four cyclic pieces

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In plain words

The four regions tile the original quadrilateral without overlap.

PG\parallel BC,\quad PFCG,GDHP\text{ cyclic}

Detailed analysis

Choose $G\in CD$ with $PG\parallel BC$ . Since $ABCD$ is cyclic, $\angle C=\pi-\angle A$ ; hence $PFCG$ is an isosceles trapezoid. Also $\angle PHD=\angle A$ and $\angle DGP=\angle C=\pi-\angle A$ , so $GDHP$ is cyclic. Thus all four pieces are cyclic.