How was greg able to say that the sides of the triangle are all equal? I know this is a regular polygon but how can we infer the sides of the triangle inside are all equal?

You can prove triangles PUT, TSR and RQP to be congruent:

PU = TS = RQ (sides of a regular polygon)

UT = SR = QP (sides of a regular polygon)

PUT = TSR = RQP (angles of a regular polygon)

Through that you can get:

PT = TR = RP

Thus, triangle PTR is equilateral