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 2 of 4: Build the first two pieces

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In plain words

The choice of $P$ in the overlapping angle region keeps all points on the intended sides.

PH\parallel AB,\quad AEPH\text{ is isosceles}

Detailed analysis

Relabel so $\angle A$ is smallest. Choose an interior $P$ , draw $PH\parallel AB$ with $H\in DA$ , and choose $E\in AB$ so $AEPH$ is an isosceles trapezoid. Choose $F\in BC$ with $\angle PFB=\angle A$ ; since $\angle PEB=\pi-\angle A$ , $EBFP$ is cyclic.