Proof: Because ABC is an isosceles right triangle.

So the angle BAC=90 degrees AB=AC.

So angle B= angle C=45 degrees.

Because AD is the center line on the side of BC.

So BD=CD=AD AD vertical BC AD is the bisector of angle BAC.

So angle BAD= angle CAD=45 degrees

Because the angle BAC=90 degrees

So AC vertical AB

Because PE is perpendicular to AB.

The angle PE is parallel to AC.

So angel ·AFE = angel ·PEF

Because PF is vertical AC, AB is vertical AC.

So PF is parallel to AB

So angel ·AEF = angel ·PFE

Because PE=PE

So triangular AEF and triangular PFE are congruent (ASA)

So AE=PF

Because PF is perpendicular to AC

So the angle PFC=90 degrees

Because angle C=45 degrees

Because angle PFC+ angle FPC+ angle C= 180 degrees.

So FPC angle =45 degrees.

So angle FPC= angle C.

So PF=FC

So AE=CF

Because angle DAE= angle C=45 degrees (proved)

AD=CD (authentication)

So triangle AED and triangle CFD are congruent.

So angle ADE= angle CDF DE=DF

Because AD is perpendicular to BC (authentication)

So angle ADC= angle ADF+ angle CDF=90 degrees.

So angle ADE+ angle ADF=90 degrees.

Because angle EDF= angle ADE+ angle ADF

So the angle EDF=90 degrees

So the triangle DEF is an isosceles right triangle.