Directly Similar Figures
What is that about?
A Mathematical Droodle

Explanation

The applet suggests the following statment:

Given two similar triangles ABC and A'B'C' and a point P, let A₁ = P + AA', B₁ = P + BB', and C₁ = P + CC'. Then triangle A₁B₁C₁ is similar to ABC and A'B'C'.

Theorem of directly similar triangles, second variant

The statement implies something more general, akin to the Fundamental Theorem of Directly Similar Figures and relates to the properties of spiral similarities. For any two directly similar figures S and S', if A is a generic point of S and A' is the corresponding point of S', then for any point P, P + AA' traces a figure similar two both S and S'.

The proof (in complex numbers) follows easily from

with m = 1 and l = -1.

References

I. M. Yaglom, Geometric Transformations II, MAA, 1968

72099606

Directly Similar Figures What is that about? A Mathematical Droodle

Explanation

References

Directly Similar Figures
What is that about?
A Mathematical Droodle