Interesting geometry result

While working on my software for 3D spline curves I needed to find a point between two others which was not halfway.

This turned into finding the ratio which an angle bisector of a triangle splits the opposite side. I worked it out with coordinate geometry and vectors, messy, messy, and then found out that this was a regular theorem in geometry. Here is the geometrical proof which I came up with. It sure ties a few things together: