Special Cases of Apollonius' Problem

In Euclidean geometry, Apollonius' problem is to construct all the circles that are tangent to three given circles. Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to:

1. three points (denoted PPP, generally 1 solution)
2. three lines (denoted LLL, generally 4 solutions)
3. one line and two points (denoted LPP, generally 2 solutions)
4. two lines and a point (denoted LLP, generally 2 solutions)
5. one circle and two points (denoted CPP, generally 2 solutions)
6. one circle, one line, and a point (denoted CLP, generally 4 solutions)
7. two circles and a point (denoted CCP, generally 4 solutions)
8. one circle and two lines (denoted CLL, generally 8 solutions)
9. two circles and a line (denoted CCL, generally 8 solutions)

In a different type of limiting case, the three given geometrical elements may have a special arrangement, such as constructing a circle tangent to two parallel lines and one circle.