[1] C is a fixed point on the axis.

[2] The cross-ratio k = (C,O

[3] The projectivity fixing points {I,J,C} and mapping G to H is a perspectivity with homology coefficient k and maps the parabola to the circle.

The properties follow from the first one, which is an easy calculation. In fact if D(0,d) the center of the circle, E(x,0), then P(x,y) is of the form (parabola) y = s-a*x

From this follows the constancy of k and this implies the third property by the very definition of the perspectivity (see Perspectivity.html ).

Perspectivity.html

Projectivity.html

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