CircumparabolaGeneration

1. Drawing parallels from vertices

In this section I examine a kind of generation of a circumparabola connected with bundles of parallel lines. I start with an exercise on some line-intersections:
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,A₀C₀} of the created parallelograms {A''B''C''B, CB''AB₀, B₀A₀BC₀} intersect at a point (see (2) of MenelausApp.html ). This implies that triangles {A''B''C'', A₀B₀C₀} have pairs of sides intersecting along the line AC i.e. they are line-perspective. By Desargue's theorem they are also point-perspective. P is the center of this perspectivity.

2. Circumparabola generation

In the previous exercise fix the direction OX and let OY describe all possible directions of the plane. Then the corresponding point P describes the parabola passing through the vertices of triangle A'B'C' and having its axis parallel to OX. Here {A',B',C'} are the middles of the sides of ABC.

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.

Remark-1 Letting also direction OX obtain all possible values we get all parabolas circumscribed on triangle A'B'C'. This is discussed in AllParabolasCircumscribed.html .
Remark-2 This method to generate a circumparabola can be very easily generalized to arbitrary circumconics which are also tangent to a given line. This is discussed in CircumconicsTangentGeneration.html .

1. Drawing parallels from vertices

2. Circumparabola generation

See Also