AE=CD, angle C= angle BAC = 60, and AB=AC can prove that triangle ADC and triangle ABE are congruent.

So angle ABE= angle DAC

And because angle DAC+ angle BAD= angle BAC=60.

So angle ABE+ angle BAD=60.

And because the horn ape = horn Abe+horn bad.

So angle ape = 60 = angle BPD.

Because the angle BDP = 90

So BP=2PQ