Fundamental Theorem of Directly Similar Figures

Let $F_0$ and $F_1$ denote two directly similar figures in the plane, where $P_1\in F_1$ corresponds to $P_0\in F_0$ under the given similarity. Let $r\in(0,1)$, and define $F_r=\{(1-r)P_0+rP_1: P_0\in F_0, P_1\in F_1\}$. Then $F_r$ is also directly similar to $F_0$.

See also Finsler-Hadwiger Theorem

References

Detemple, D. and Harold, S. ``A Round-Up of Square Problems.'' Math. Mag. 69, 15-27, 1996.

Eves, H. Solution to Problem E521. Amer. Math. Monthly 50, 64, 1943.