Problem 5
Two tangent circles and a point P on their common tangent perpendicular to the line joining their centres are given. Construct with ruler and compass all circles tangent to the two given circles and passing through P.
Step 2 of 5: Show that each given circle is invariant
Detailed analysis
Any secant through P cuts a given circle at X,Y, and the power of P is PX times PY=PT squared. Thus inversion swaps X and Y on that circle, so it maps each given circle to itself.