# Directly Similar Figures

What is that about?

A Mathematical Droodle

|Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

### Explanation

The applet suggests the following statment:

Given two similar triangles ABC and A'B'C' and a point P, let A_{1} = P + AA', B_{1} = P + BB', and C_{1} = P + CC'. Then triangle A_{1}B_{1}C_{1} is similar to ABC and A'B'C'.

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,

The proof (in complex numbers) follows easily from

with m = 1 and l = -1.

### References

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

|Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

64479875 |