Inversion preserves orthogonal circles
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In the picture above, the two circles intersect orthogonally at the points A and B. The point P' is the inverse of P with respect to the blue circle. As the point P moves around the black circle, the inverse P' also moves around the black circle.
You can move the points A and B to resize the black circle and move the points O and X to resize the blue circle.
Exercise. Given two points A and B on a circle, can you construct the unique circle which intersects orthogonally at A and B? Click here for the inversive geometry interactive website.