ParabolaNewton

Newton's generation of parabola

Let L be a fixed line (x-axis), F a fixed point not on L and u=angle(COE) a fixed angle. For each point X on L construct an angle(FXY') = u. Then side XY' of the angle envelopes a parabola as X moves on L. The focus of the parabola is F and its vertex is easily constructed from the given data.

The claim reduces to the construction of Thales Parabola as this is described in ThalesParabola.html .
In fact, let Y be the projection of F on the side XY' of the variable angle. The circumcircle of triangle XYF passes through a fixed point B, which is the projection of F on L. Constructing at B the angle u determines the fixed line L' on which moves vertex Y for all locations of the point X on L. Thus, lines XY are defined through the intersections of the two fixed lines {L,L'} with all members of the bundle of circles passing through the fixed points {F,B}. By the arguments in the aforementioned reference follows that the vertex of the parabola is the projection A of F on line L'.

Newton's generation of parabola

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