Petersen-Shoute Theorem

1. If $\Delta ABC$ and $\Delta A'B'C'$ are two directly similar triangles, while $\Delta AA'A''$, $\Delta BB'B''$, and $\Delta CC'C''$ are three directly similar triangles, then $\Delta A''B''C''$ is directly similar to $\Delta ABC$.

2. When all the points $P$ on $AB$ are related by a Similarity Transformation to all the points $P'$ on $A'B'$, the points dividing the segment $PP'$ in a given ratio are distant and collinear, or else they coincide.

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 95-100, 1967.