Circle of Similitude

Circle of Similitude: What Is It?
A Mathematical Droodle

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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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The circle C_S 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 C_S. 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,

d_i² = t_i² + R_i²,

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 C_S the two circles are seen under equal angles. This follows from the similarity of triangles with side lengths t_i, d_i, and R_i, 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 C_S.

Circle of similitude has additional engaging and unexpected properties.

References

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

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