The simple answer is that the method of proof is not the simplest, but this method I wrote is more convenient.

D is the vertical line between AC and AB, and the vertical foot is L and K..

There is KD=LD,

Let KD and PE intersect at point H.

DH=HP (because △DPH is Rt△, similar to Rt△ABC)

DH=LE, PH=KF, angle DLE= angle DKF=90 degrees.

Rt△LDE is equal to Rt△KDF.

Therefore, DE=DF and angle LDE= angle FDK.

In addition, angle LDE+ angle EDK=90 degrees, so angle EDF= angle FDK+ angle EDK=90 degrees, which is proof.