In this case, it's all about how triangles behave on a unit sphere, if one edge gets close to the length of π, or half a circumference. For Earth and Miles, r is 3963 and r * π = 12450, which is awfully close to 12446.
We are effectively looking at a https://en.wikipedia.org/wiki/Spherical_lune here. The dihedral angle can be chosen freely. One half great circle is going directly between the antipodal points, while the other half great circle is intersected at the ratio into two edges.
So all you need to find are two antipodal points. Then any point lying on the two "small" circles defined by the ratio in either direction of the half great circle fullfills this condition. Helpful if you have bit of wiggle room with a place like Nazca.
If we take for simplicity the North and South Pole then any point at the latitude 21.25 North or South would fullfill this condition. Mecca at 21.4N would come within 15 km of that band already.
