My two previous puzzles have been based on the nine-point circle of a triangle. They both stemmed from a series of equations I worked out for the angles between the nine-point centre and each of the nine points. The following is based on an acute triangle, as has been shown you can swap a vertex with the orthocentre to get to that situation. Every third point on the circle is the midpoint between a vertex and the orthocentre, and the order you encounter midpoints / altitude feet along each side depends only on the order of the angle sizes.

As you can see, at least four of those angles will be the same. In puzzle 359, the apex angle was the lowest angle ‘c’ and ‘a’ and ‘b’ were equal. In puzzle 360, it was ‘b’ and ‘c’ that were equal and ‘a’ was at the apex.