Homologous Lines under Three Spiral Similarities
The following result is classic in the theory of three directly similar figures, see [Casey, Johnson, Lachlan, Yaglom].
Let F_{1}, F_{2}, F_{3} be any three figures which are directly similar; let O_{1} be the center of similitude of F_{2} and F_{3}; O_{2} that of F_{3} and F_{1}; and O_{3} that of F_{l} and F_{2}. The triangle formed by the three centers of similitude O_{1}, O_{2}, O_{3} is called the triangle of similitude of the figures F_{1}, F_{2}, F_{3}; and the circumcircle of this triangle is called the circle of similitude. There is a good reason for this designation.
Theorem
In every system of three directly similar figures, the triangle formed by three homologous lines is in perspective with the triangle of similitude, and the locus of the center of perspective is the circle of similitude. |
Two directly similar figures can always be obtained from each other by either rotation, homothety, translation, or, more generally, spiral similarity. The latter is a combination of rotation and homothety with the same center. It is just characterized by an angle (of rotation) and by a real number (the coefficient of homothety.)
Since the theorem deals with homologous (corresponding) line segments, in the applet below three similar figures are represented by the three segments A_{1}B_{1}, A_{2}B_{2}, A_{3}B_{3}. The center of the spiral similarity that maps A_{2}B_{2} on A_{3}B_{3} is O_{1}, while O_{2} and O_{3} are the centers of transforms that map A_{3}B_{3} on A_{1}B_{1} and A_{1}B_{1} on A_{2}B_{2}, respectively.
In the applet, the three segments can be dragged by their endpoints, causing a rotation around their other end or stretching their lengths, or by their midpoint, causing a translation. Checking the "construction" box shows the three pairs of circles that spiral similarity intersect at the centers of the three spiral similarities.
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 BogomolnyProof
We shall prove a little more than stated in the theorem.
First of all, note that we are in a position to apply the Pivot Theorem which tells us that the circles D_{1}A_{2}A_{3}, A_{1}A_{2}D_{3} and A_{1}D_{2}A_{3} intersect in a point (Hint A). Call it V. By the same token, circles D_{1}B_{2}B_{3}, B_{1}B_{2}D_{3} and B_{1}D_{2}B_{3} also intersect in a point (Hint B). Call it W.
What if applet does not run? |
Observe that by the construction of centers of similarity circles D_{1}A_{2}A_{3} and O_{1}A_{2}A_{3} coincide as are the circles A_{1}A_{2}D_{3}, A_{1}A_{2}O_{3} and A_{1}D_{2}A_{3}, A_{1}O_{2}A_{3}. A similar assertion holds for the points B as well.
Let U be the intersection of lines D_{2}O_{2} and D_{3}O_{3}. We'll show that the four points U, V, O_{2}, O_{3} are concyclic. Since proving that U, V, O_{1}, O_{3} are also concyclic is similar, it will then follow that the points O_{1} and O_{2} lie on the circle UVO_{3} so that all five of them -
U, V, O_{1}, O_{2}, O_{3} - are concyclic. Since V and W enter the problem in a symmetric fashion, W lies on the same circle. Furthermore, the same argument would have led to the same conclusion had we taken U to be the intersection
So, now everything hinges on proving that
we see that
(1) | ∠O_{2}VO_{3} + ∠D_{2}O_{2}V + ∠D_{3}O_{3}V + ∠O_{2}UO_{3} = 360°. |
Since points D_{2}, O_{2}, V, A_{1} are concyclic,
∠D_{2}O_{2}V = 180° - ∠D_{2}A_{1}V. |
Since points D_{3}, O_{3}, V, A_{1} are concyclic,
∠D_{3}O_{3}V = ∠D_{3}A_{1}V. |
Taking into account that ∠D_{2}A_{1}V = ∠D_{3}A_{1}V,
∠O_{2}VO_{3} + ∠O_{2}UO_{3} = 180°, |
implying that points U, O_{2}, V, O_{3} are concyclic, as required.
Note that the derivation relied heavily on the specifics of the point layout depicted in the diagram. For example, (1) was true because quadrilateral O_{2}VO_{3}U was convex. It need not be so. Had we set to prove that points U, O_{1}, V, O_{3} were concyclic, we would have to proceed differently since, in the same diagram, quadrilateral O_{1}VO_{3}U is not convex (not even simple.) The ambiguity as well as the reliance on the diagram may have been avoided had we used directed angles modulo 180°.
(There are additional points on the circle of similitude. For example, if there are two triplets of homologous lines, their triangles D_{1}D_{2}D_{3} are directly similar with the center of the spiral similarity that maps one on the other lying on the circle of similitude, naturally.)
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