Construct a circle cutting two other circles at right angles

Question

I have the following problem:

On a line $l$ on this line are the centers of two circles $C_1$ and $C_2$ . Circles $C_1$ and $C_2$ do not intersect and are not tangent to eachother. (but one could be completely inside the other)

How to construct the circle $C_3$ also with its center on line $l$ that cuts both $C_1$ and $C_2$ at right angles?

This construction is (a bit) described at https://en.wikipedia.org/wiki/Ultraparallel_theorem there the explanation is a bit to complex (because it wants to use specified hyperbolic motions) and is i think not complete.

I would like to learn a simplified straight edge and compass construction.

Answer 1

there are lots of circles that are orthogonal to both $C_1, C_2.$ they all have centers on the radical axis of $C_1, C_2.$ radical axis is line orthogonal to the center line $l$ and has tangents of equal length to $C_1, C_2$

we can construct the radical axis by drawing a circl $C$ on one side of the line(say) of $l$ cutting both circles $C_1, C_2.$ now, find the points where $C, C_1$ cut and draw the line. similarly, draw the line at which $C, C_2$ cut. these two lines intersect on the radical axis.

pick any point $P$ on the radical axis of $C_1, C_2$ and draw the two equal tangents to $C_1, C_2.$ find the contact points $T_1, T_2.$ draw a circle with center $P$ and radius equal $PT_1.$ this circle will be orthogonal to both.

do the same on the other side of $l.$ now you have two points on the radical axis. draw any circle with center on this radical axis, and it must cut $C_1, C_2$ orthogonally.