Problem of Apollonius

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Apollonius' problem is to find a circle that is tangential to a given set of three circles. The three fixed circles can be of any size or distance to one another. There are eight possible solutions, as shown in the Figure.

This problem can be solved using circle inversion. We first increase (or decrease) the radii of all three given circles by the same amount until two of the circles are exactly tangential. Inversion about their tangent point transforms the two touching circles into two parallel lines. (They're parallel because the intersection point at the origin is transformed into a point at infinity under inversion and parallel lines intersect only at infinity.) The same inversion transforms the third circle into another circle. We then construct a circle tangential to the two parallel lines that touches the third circle; re-inversion produces the desired circle.

[edit] References

• C. Stanley Ogilvy (1990) Excursions in Geometry, Dover. ISBN 0-486-26530-7.