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Four Touching Circles: What Is this About?
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(The circled points can be dragged and also the gray circle.)
Copyright © 1996-2008 Alexander Bogomolny
Four Touching Circles
The applet suggests the following theorem:
| (A) | Let there be 4 circles S1, S2, S3, and S4 each tangent cyclically to its neighbors, so that S1 touches S2 and S4, S2 also touches S3, and the latter touches S4. Prove that the four tangency points are concyclic, i.e. lie on a circle. |
Check the "Inverted diagram" box. You'll see two parallel lines, two tangent circles, each of which is tangent to one of the parallel lines and a line that goes through the three points of tangency. What is this about?
Assume we indeed have a chain of four circles, each touching two of its neighbors. Make an inversion with one point of tangency as the center. Two circles tangent at that point will map on two parallel lines. The other two circles map onto tangent circles each touching one of the parallel lines. (Because of the angle preservation property tangent curves are mapped onto tangent curves.) Observe that
Two parallel lines are homothetic with any point in the plane (but not on the lines) as a legitimate center of homothety.
Two tangent circles are homothetic with the common point of tangency as the center.
This means that that the whole configuration is homothetic in the point of tangency of the two circles. Necessarily the points of tangency of the circles with the parallel lines are images of each other under that homothety, such that the three points of tangency are collinear. By the inversion at hand, the line those points are on, is mapped onto a circle passing through the center of inversion.
The proof may now proceed as follows. We used 1 of the points of tangency of the circles S1, S2, S3, and S4 as the center of inversion. The other three points are of course concyclic. Let them lie on circle S. Under the inversion, the images of the three points are collinear. Therefore that circle S passes through the center of inversion - the fourth point of tangency.
Remark
This statement has a more elementary proof. The points of tangency lie on the lines joining the centers of adjacent circles. The quadrilateral formed by the centers has a property (the sums of the two pairs of opposite sides are equal) that makes it inscriptible. The latter property has been actually proven (Annales de Gergonne, tome VI, 1815-1816, p. 46) by J.-B. Durbande [F. G.-M., note no 745, p. 317].
References
- F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, pp. 1175-1177
Inversion
- Angle Preservation Property
- Apollonian Circles Theorem
- Archimedes' Twin Circles and a Brother
- Bisectal Circle
- Chain of Inscribed Circles
- Circle Inscribed in a Circular Segment
- Circle Inversion Tool
- Feuerbach's Theorem: a Proof
- Four Touching Circles
- Hart's Inversor
- Inversion in the Incircle
- Inversion: Reflection in a Circle
- Inversion with a Negative Power
- Peaucellier Linkage
- Polar Circle
- Poles and Polars
- Radical Axis of Circles Inscribed in a Circular Segment
- Steiner's porism
- Three Tangents, Three Secants
Copyright © 1996-2008 Alexander Bogomolny
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