Or search by topic
Congratulations to Tony from State College Area High School, Pennsylvania, USA for this solution.
First of all, here is the solution to finding the equation of the orthogonal circle for the circles with centers of $(0,0)$, $(3,0)$, $(9,2)$ and radii respectively of $5$, $4$, and $6$.
As the circles are orthogonal we can draw three right angled triangles. One of the legs of each right triangle is the radius of one of the given circles, the other leg is the radius of the unknown orthogonal circle, and the hypotenuse is the distance between the center of the known circle and the center of the unknown orthogonal circle.