Given a triangle ABC and two independent directions {OX, OY} draw from the vertices of the triangle lines parallel to the two directions. These parallels create three parallelograms having the sides of the given triangle as diagonals. The other three diagonals of these parallelograms pass trhough one point P.

The diagonals {A''C'',AC,A

A proof results by showing that P describes a conic and identifying the conic with the claimed parabola. This can be easily done using {A',B',C',D'} as a projective basis (D' being the centroid of A'B'C') and calculating the intersection point P of lines in this coordinate system. The resulting equation is a quadratic one.

Then it is also easy to see that for appropriate positions of OY point P obtains the positions {A',B',C',A'',B'',C''}, where the three last points are the intersections of lines parallel to OX from {A,B,C} with the sides of ABC.

Using the results of AnticomplementaryAndCircumparabola.html , the previous calculations show that the conic coincides with the parabola claimed.

AnticomplementaryAndCircumparabola.html

CircumconicsTangentGeneration.html

MenelausApp.html

Parabola.html

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