An almost real life geometry problem

I needed to move a point around a circle, in a computer graphics application, using the mouse pointer. It is clearly not sensible to have mouse pointer on the point all the time, so the problem was

“For a point anywhere, where is the point both on the circle and on the radial line?”

point on circle 2

It may help to see the situation without the coordinate grid on show:

point on circle 1

This is a problem with many ways to a solution, some of them incredibly messy !

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