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! Continue reading A Solution for the Mathematically Inclined →

I’ve been thinking about the following math problem lately (I occasionally get distracted by such things):

Suppose you have four points in 3D space, a, b, c, and p, where a, b, and c are distinct and non-colinear and their positions are known, and p is the unique point defined by the following:

1. 
2. 
3. 
4. p lies in the interior of the triangular prism defined by a, b, and c that is perpendicular to the plane containing a, b, and c.

As an image:

If there is only one point satisfying the above criteria, can it be found without knowing γ?  I plan on posting my proof next week Thursday, 2016-02-25.