<Polygon-2023-259>, hint

Let AB=AC=b, BC=a, <ABC=<ACB=Q.

Triangle DPC is isosceles, DP=PC=b-a, <DPC=Q/2.

MKHI is parallelogram ( KM=//MI), KH=HI+ME, ME=b-(b-a)=a

Triangle EFM is isosceles. <MPE=Q/2, <MPE+<PME=<PEG=Q.

Hence M,P,C are collinear.