Circles of Apollonius

One of the first questions of enumerative geometry was posed by Apollonius

How many plane circles are there that touch three given circles (tangentially)?

Solution: There are 8 possible circles that meet this condition:

The solution circle touches …

(1) all three circles $C_{1}, C_{3}, C_{3}$ on the outside,
(2) the circle $C_{1}$ in the inside, the circles $C_{2},C_{3}$ on the outside,
(3) the circle $C_{2}$ in the inside, the circles $C_{1},C_{3}$ on the outside,
(4) the circle $C_{3}$ in the inside, the circles $C_{1},C_{2}$ on the outside,
(5) the circles $C_{2},C_{3}$ in the inside, the circle $C_{1}$ on the outside,
(6) the circles $C_{1},C_{3}$ in the inside, the circle $C_{2}$ on the outside,
(7) the circles $C_{1},C_{2}$ in the inside, the circle $C_{3}$ on the outside,
(8) all three circles $C_{1}, C_{3}, C_{3}$ on the inside.

Due to Heinz Klaus Strick