### Pairs of Homologous Lines under Spiral Similarities

One of the most basic theorems of the theory of three directly similar figures (see [Casey, Johnson, Lachlan, Yaglom]) claims the existence of a circle of similarity of the three figures that contains a variety of special points. The applet below illustrates an extension of this result:

Let F |

In the applet, two centers of similarity can be dragged and their rotation angles and coefficients modified with the dials on the left side of the applet. ΔA_{1}B_{1}C_{1} can also be dragged either as a whole or modified by dragging its vertices.

What if applet does not run? |

### References

- J. Casey,
*A Sequel to the First Six Books of the Elements of Euclid*, University of Michigan, 2005 (reprint of 1888 edition), pp. 189-193 - R. A. Johnson,
*Advanced Euclidean Geometry*, Dover, 2007 (reprint of 1929 edition), pp. 302-312 - R. Lachlan,
*An Elementary Treatise on Modern Pure Geometry*, Cornell University Library (reprint of 1893 edition), pp. 140-142 - I. M. Yaglom,
*Geometric Transformations II*, MAA, 1962, p. 82, pp. 163-165

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Copyright © 1996-2018 Alexander Bogomolny### Proof

That triangles D_{1}D_{2}D_{3} and D'_{1}D'_{2}D'_{3} are perspective to ΔO_{1}O_{2}O_{3} from points on the circle of similitude has been proved elsewhere. We only have to show that they are directly similar and that the center of spiral similarity that maps one on the other is located on the circle of similitude.

What if applet does not run? |

Indeed, by the construction, the side lines of triangles D_{1}D_{2}D_{3} and D'_{1}D'_{2}D'_{3} are obtained from each other in pairs by fixed rotations. It follows that the corresponding lines in the two triangles are equally inclined to eacher other so that the triangles are similar.

We'll have to consider to separate cases: (a) the triangles have three parallel sides, (b) not all sides are parallel.

... to be continued ...