Problem of Apollonius

Animation of the solution to Apollonius' problem using the method of inversion. The original three circles (red, green and blue) are first expanded by the same amount ε until two of them (the red and the blue) touch. A grey circle of inversion is centered on their point of tengency, such that it intersects both the red and blue circles in two points. Upon inversion, the red and blue circles become parallel red and blue lines, whereas the green circle becomes another circle. There are several positions at which a yellow circle can be tangent to the two lines and to the new green circle. Upon re-inversion and shrinking its radius by ε, these yellow circles should be tangent to all three of the original three circles.

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 above figure.

Solution by inversion

Apollonius' 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.

Trivia

Eight circles that touch three fixed circles (shown in black)

The Desborough Mirror, a beautiful bronze mirror made during the Iron Age between 50 BC and 50 AD, consists of arcs of circles that are exactly tangent.

References

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