Circles C and C' both touch the line AB at B. Show how to construct all possible circles which touch C and C' and pass through A.
Solution
Take a common tangent touching C' at Q' and C at Q. Let the line from Q to A meet C again at P. Let the line from Q' to A meet C' again at P'. Let the C have center O and C' have center O'. Let the lines OP and O'P' meet at X. Take X as the center of the required circle. There are two common tangents, so this gives two circles, one enclosing C and C' and one not.
To see that this construction works, invert wrt the circle on center A through B. C and C' go to themselves under the inversion. The common tangent goes to a circle through A touching C and C'. Hence the point at which it touches C must be P and the point at which it touches C' must be P'.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002