This is Feuerbachs circle or Eulers circle or nine point circle. M is the center of the circumcircle of the podic triangle. Just google it: nine point circle
I'm asked to prove any conjectures about the constructed triangle and the points A' B' C' M1 M2 M3 N1 N2 N3. A few details:
given triangle ABC, A' B' C' are altitudes with the orthocenter P
M1 M2 M3 are midpoints of AP BP and CP
N1 N2 and N3 are the midpoints of AB BC and AC
G is the circumcenter
M is the midpoint of PG
and there is a circle with center M and radius MA'
Ok so obviously A' B' C' and N1 N2 and N3 are all radiuses, so they are equidistant from M, but what is M? is that just the general center of ABC? what kind of special center is M for the triangle? also m1 m2 and m3 are radiuses. so is the whole thing to prove that they are equidistant? I want to think that there is more to this long problem than just that the points all lie on the same circle. it seems too simple.