Suppose we are a creative individual, and during our math exam would like to draw a picture of a cube using the vanishing point perspective. Let $A$ and $B$ be two adjacent vertices of a square in the plane, and $V$ the vanishing point. We extend segments from each vertex of the square to $V$.
The question is how far along each segment should we place the remaining vertices of the cube. Let $E$ be the remaining vertex lying on the segment $\overline{BV}$:
Proposition: $E$ is the intersection of the bisector of $\angle VAB$ and the segment $\overline{BV}$, so that $\angle VAE=\angle EAB$.
Proof: In the plane, the bisectors of opposite vertices of any square should be equal. If this fact holds in any projected square, this gives a unique solution and we are done.
Can anyone provide a proof for this claim?
