Circle of Similitude: What Is It?
A Mathematical Droodle


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Explanation

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Two circles of different radii that lie without each other possess two centers of similarity. These lie at the intersection of their common internal and external tangents. With the reference to the applet below, the circles c1, c2 are given by their centers C1, C2 and radii R1, R2. The centers of similarity are denoted P and Q.


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What if applet does not run?

The circle CS constructed on PQ as diameter is known as the Circle of Similitude of the two circles; and for a good reason, too. Let T be any point on CS. Introduce t1, t2, the length of tangents from T to c1, c2. Let d1, d2 be the distances from T to the centers C1, C2. Then

  t1/t2 = d1/d2 = R1/R2.

As we know, the locus of points whose distances are in a fixed ratio to two given points is constant is an Apollonian circle with respect to the two points. For the two points C1, C2 and the ratio R1/R2, this is exactly the circle of similitude defined above. Thus for this circle

  d1/d2 = R1/R2.

is automatic. However, by the Pythagorean theorem, for i = 1, 2,

  di2 = ti2 + Ri2,

which implies

  t1/t2 = R1/R2.

The segment of a tangent from a point to a circle between the point and the point of tangency is known as tangential segment. What we just showed is that the lengths of the tangential segments from a point on the circle of similitude of two circles is in the same ratio as the circles' radii as that of the distances from the point to the circles' centers.

As hinted by the applet, the points on the circle of the similitude possess another property: from any point T on CS the two circles are seen under equal angles. This follows from the similarity of triangles with side lengths ti, di, and Ri, i = 1, 2. The two circles are obtained from each other by homotheties centered at P and Q, or by a (more general) spiral similarity with center at any other point on CS.

Circle of similitude has additional engaging and unexpected properties.

References

  1. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
  2. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960

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Copyright © 1996-2017 Alexander Bogomolny

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