Circle of Similitude: What Is It?
A Mathematical Droodle
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Copyright © 1996-2018 Alexander BogomolnyTwo 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 c_{1}, c_{2} are given by their centers C_{1}, C_{2} and radii R_{1}, R_{2}. The centers of similarity are denoted P and Q.
What if applet does not run? |
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 t_{1}, t_{2}, the length of tangents from T to c_{1}, c_{2}. Let d_{1}, d_{2} be the distances from T to the centers C_{1}, C_{2}. Then
t_{1}/t_{2} = d_{1}/d_{2} = R_{1}/R_{2}. |
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 C_{1}, C_{2} and the ratio R_{1}/R_{2}, this is exactly the circle of similitude defined above. Thus for this circle
d_{1}/d_{2} = R_{1}/R_{2}. |
is automatic. However, by the Pythagorean theorem, for
d_{i}^{2} = t_{i}^{2} + R_{i}^{2}, |
which implies
t_{1}/t_{2} = R_{1}/R_{2}. |
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},
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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