Cabri has a built in method for constructing the unique conic through five points.

focus, directrix and point Classically, a conic C is defined in terms of a line D, the directrix, a point F, the focus, and a positive number e, the eccentricity. In fact, C is the locus

C={P : |PF|=e|PD|},

where |PD| denotes the distance of the point P from the line D.

This construction can be implemented in Cabri, but it is easier to define the eccentricity implicitly by giving a particular point on the conic.

parabola For a parabola, the eccentricity is 1, so we can specify it by just the focus and directrix.
central conic Just as there is a unique conic through five points, there is a unique central conic (ellipse or hyperbola) through three points.
five lines Just as there is a unique conic through five points (with no three in a line), there is a unique conic touching five lines (with no three concurrent or parallel). This is of course the dual property.

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