Make an arc with A and B as the center and more than half the length of AB as the radius.

Get two intersections c, d

Connect CD and AB in O, then O is the midpoint of AB.

Make an arc with O as the center and OA as the radius, and intersect CD at point E.

Connecting AE

AE=√(AO？ +OE？ )=√2AO=√2AB/2

∴AE:AB= 1:√2

Next, make an arc with A as the center, AE as the radius, and the intersection with AB is the point P you want.