Derive a condition for the two circles

\[\begin{align*} x^2+y^2+2g_1x+2f_1y+c_1 &= 0,\\ x^2+y^2+2g_2x+2f_2y+c_2 &= 0, \end{align*}\]

to cut orthogonally.

‘Orthogonally’ means ‘at right angles’.

Could we draw a helpful diagram? If we assume two circles cut orthogonally, can we deduce a relationship between their radii and the distance apart of their centres?

Hence, or otherwise, show that in general there is just one circle orthogonal to three given circles.

If such a circle exists, what equations can we write down? When do these equations have a unique solution?