<Tri-2024-2125>, hint

Actually M=O.

D,E,F are Symmetric to O wrt BC, AC,AB respectively.

In triangle AHE  AH=2R cosA, AE=R ( radius of circumcircle of triangle ABC)

<HAC=90-C, <EAC=90-B, Hence <HAE=A.

Apply cosine 2nd rule to AHE,

HE^2=AE^2+AH^2-2AE xAH cosA =R^2+(2R cosA)^2-2R (2R cosA) cosA

Hence HE=R. HE=HF=HD=R.