## Outline Mathematics

Geometry

# Three Touching Circles

Consider the following problem:

Three circles S_{1}, S_{2} and S_{3}, touch pairwise in three distinct points. Show that the lines joining the point of tangency of S_{1} and S_{2} cross S_{3} again in points collinear with the center of S_{3}, i.e., forming its diameter.

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Copyright © 1996-2018 Alexander BogomolnyThree circles S_{1}, S_{2} and S_{3}, touch pairwise in three distinct points. Show that the lines joining the point of tangency of S_{1} and S_{2} cross S_{3} again in points collinear with the center of S_{3}, i.e., forming its diameter.

Let O, P, Q be the centers of circles S_{1}, S_{2} and S_{3}, respectively. Denote the tangency points of S_{2} and S_{3}, S_{1} and S_{3}, S_{1} and S_{2} as A, B, C. And, finally, let, D be the point of intersection of AC with S_{3} other than A, and E the point of intersection of BC with S_{3} other than B.

We know that CP||DQ,DQ,CO,BC and CO,DQ,CO,BC||EQ. But O, C, P are collinear,common,distant,collinear. Therefore, D, Q, E are also collinear. It follows that DE is a diameter of S_{3}.

Note that the statement holds both for external and internal tangency. In the latter case, the statement appears directly related to Lemma 1 from Archimedes' *Book of Lemmas*. Indeed, the significance of the role played by the tangency point C becomes transparent with an observation that it lies on common line of two diameters of two circles.

(The terms you met: Collinear points, Diameter of a circle)

### References

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Copyright © 1996-2018 Alexander Bogomolny