The answer to my post last week is yes. Here’s the quick (intuitive) proof:
First, note that and is the surface of two spheres in three dimensions, and their intersection is either a circle or a point (we know they are not the same sphere since a and b are distinct and that they must intersect because we know p exists). If it is a point, then p is that point (and since we have not used gamma we have proved the hypothesis).
If the intersection is a circle, it intersects the plane containing a, b, and c at two points, but only one is inside the triangular prism, and that must be the point p. We know it cannot be a point not on the plane, since if it was, two points would satisfy , and not satisfy the uniqueness condition.
I personally prefer algebraic proofs though, so let’s crank some algebra!
It is clear that the triangular prism does not provide any constraints on p3, so the uniqueness condition enforces that p3=0. Thus, resolving for p1 provides:
Since this point must lie within the triangle a b c, p1 must have the same sign as c, so: