Parabola Constructions

Solutions for all the parabola challenges can be found at the links below.

Parabola 1
Parabola 2
Parabola 3
Parabola 4
Parabola 5
Parabola 6
Parabola 7
Parabola 8
Parabola 9
Parabola 10
Parabola 11
Parabola 12

One basic property of parabolas ended up being the objective in every one of these constructions. From any point on the curve the distance to the focus is equal to the distance to the directrix. Once those two objects were produced, the construction was essentially complete. Here the focus, F, and the directrix are given.

Point T is what I call a traveling point. It is attached to an object, and its motion along that path is what drives the locus. In this case T is attached to the directrix. Through T construct a line perpendicular to the directrix. Construct the perpendicular bisector of FT. These two lines intersect at U, which is on the curve, and the parabola is the locus of U as T travels along the directrix.

For a complete parabola construction, I would expect the curve itself, the focus, the axis, the axial vertex, and the directrix. For example, in the case above, the axis and vertex constructions are actually very simple. I do not intend to go into details like that in these construction documentations.

Under certain given conditions the required curve might not be possible. In those cases, someone with dynamic geometry chops should still be able to pass the drag test. When the given objects are dragged into positions that do not allow a solution, all constructed objects should vanish. I do not intend to explain all of those details either.

The parabola is the simplest of the three classes of conic sections, so it was surprising to find that these challenges resulted in some of the most complicated solutions.

Back to Conic Challenges