# Two Triples of Similar Triangles

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A Mathematical Droodle

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21 December 2016, Created with GeoGebra

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Copyright © 1996-2018 Alexander Bogomolny

What if applet does not run? -->Draw two triangles, say, A_{1}B_{1}C_{1} and A_{2}B_{2}C_{2} both similar (and similarly oriented) to triangle ABC. Choose another triangle, say, 123, and construct three triangles A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3} and C_{1}C_{2}C_{3} similar (and similarly oriented) to 123 on the segments A_{1}A_{2}, B_{1}B_{2} and C_{1}C_{2}, respectively. Then ΔA_{3}B_{3}C_{3} will be similar to ΔABC!

The configuration is completely symmetric between names and indices. One can start with two triangles similar to 123 and after joining their corresponding vertices erect triangles similar to ABC. The newly constructed vertices form then a triangle similar to triangle 123.

If triangles A_{1}B_{1}C_{1} and A_{2}B_{2}C_{2} are translations of each other, the result is immediate, and the third triangle A_{3}B_{3}C_{3} is obtained by translation from either of the first two.

Otherwise, there exists a unique spiral similarity with center O that transforms A_{1}B_{1}C_{1} into A_{2}B_{2}C_{2}. We then have

_{2}/OA

_{1}= OB

_{2}/OB

_{1}= OC

_{2}/OC

_{1}and

_{1}OA

_{2}= ∠B

_{1}OB

_{2}= ∠C

_{1}OC

_{2}.

It follows that triangles A_{1}OA_{2}, B_{1}OB_{2} and C_{1}OC_{2} are similar. But we also assumed the similarity of triangles A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3} and C_{1}C_{2}C_{3}. So that triangles A_{1}OA_{3}, B_{1}OB_{3} and C_{1}OC_{3} are similar, from where

_{3}/OA

_{1}= OB

_{3}/OB

_{1}= OC

_{3}/OC

_{1}and

_{1}OA

_{3}= ∠B

_{1}OB

_{3}= ∠C

_{1}OC

_{3}.

Therefore, ΔA_{1}B_{1}C_{1} is mapped on ΔA_{3}B_{3}C_{3} by spiral symmetry. They are, therefore, similar.

### Remark

The result we just proved is a formal consequence of the *Fundamental Theorem of Directly Similar Figures*, in which real coefficients are replaced with complex numbers. (I am grateful to Steve Gray for bringing this to my attention.)

### Reference

- H.S.M. Coxeter, S.L. Greitzer,
*Geometry Revisited*, MAA, 1967 - D. Wells,
*You Are a Mathematician*, Dover, 1970 - I. M. Yaglom,
*Geometric Transformations II*, MAA, 1968

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