Solution:

Suppose not. Let $A$ be a white point and $B$ a black point. Their midpoint $C$ is one of the two colors; without loss of generality suppose $C$ is black. Now pick any point $D$ not collinear with $A, B, C$, and construct $E$ so that $C A D E$ is a parallelogram. If $D, E$ are both white, then $C A D E$ has three white vertices and one black vertex, impossible; if they are both black, then $C A D E$ has three black vertices and one white vertex, impossible. So $D$ and $E$ are opposite colors.

But $B C D E$ is also a parallelogram, since $B C=A C=D E$ and lines $B C, D E$ are parallel. However, $B C D E$ it has three black vertices and one white vertex. Thus we have a contradiction.