<Tri-2025-2248>, hint

Since {I(A)=Ia } is A-excenter of triangle ABC IaM, IaN, IaP are the radii of 

A-excircle.  Since IaMNP is  a rhombus , IaM=PN=IaP,  hence triangles 

IaPN, IaMN are eqilateral. Thus <ABC=<PIaN=60=<NIaM=<ACB

Triangle ABC is equilateral.