It is well known that for any triangle, the following nine points:

a) the three midpoints,

b) the three feet of the altitudes (the altitudes are the lines from each vertex to the opposite side, perpendicular to that side),

c) and the three points midway between the vertices and the orthocentre (the orthocentre is the point where the three altitudes cross),

all lie on a circle, known as the Nine-Point Circle.

Sometimes some of those points coincide with one another, for instance the midpoint of the base of an isosceles triangle is also the foot of the altitude from the apex down to the base.

I have an isosceles triangle, and so since two of the nine points coincide, there are now eight distinct points. In my triangle these eight points are equally spaced, forming the vertices of a regular octagon.

What is the angle at the apex of the isosceles triangle?

For bonus points, what is a second possible value for the apex angle?