Inversion of a point

The inversion studied in this sketch is the inversion with center O and inversion-circle the green circle. You may drag the red point M. It's image in the inversion is the blue point M'. In terms of complex numbers, if z is the affix of M, the affix of M' will be 1/conj(z), that is the algebraic inverse of the complex conjugate of z.

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Problems

1. For what positions of M do we have M' = M?


2. For fixed O and M, is it possible to choose the point 1 so that M' = M?


In the following problems you should make use of the buttons.

3. If you move M along a vertical line, on what kind of curve does M' move?


4. Draw in the picture an approximative sketch of the set of points M such that M' is on the square or outside it.
Draw a cleaner picture on the paper!