<Tri-2025-2217>, hint

Consider Simson Line.

Suppose M on arc AC and let N'. P', Q' be the projections of M onto lines AB, BC,CA

then N'P'Q' are collinear, which is Simson line. Let N, P, Q symmetric to M 

about AB, BC, CA, then N,P,Q and N'.P',Q' are parallel.

Let H(A), H(B),H(C) be symmetric to H, orthocenter of ABC about BC, CA, AB

Then H(A), H(B), H(C) lie on the circumcircle of ABC.

Suppose H=M, H(A)=P, H(B)=Q, H(C)=N.

IF H=O (circumcenter) then ABC is equilateral.