Begin with triangle DBC.

Imagine the rest of the figure is not yet in place.

Next attach triangle BQD which is congruent to triangle DBC.

Then attach triangle PQD which is also congruent.

The only way for P, D, C to be collinear is if

Finally attach triangle APQ.

We want to ensure that A, Q, B are also collinear if

and also that A, P, D, C are collinear.

From triangle DBC, if A, Q, B are collinear.

Then if A, P, C, D are collinear.

From this it follows that

in order to fulfill the condition that A, P, D, C are collinear

and A, Q, B are collinear.

creates both sets of collinearities.