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.

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

Find the centre and radius of the circle orthogonal to the three circles

\[\begin{align*} x^2+y^2-12x-8y+34 &= 0,\\ x^2+y^2-6x-12y+32 &= 0,\\ x^2+y^2-10x-6y+30 &= 0. \end{align*}\]