Center of an Equilateral Triangle Circumscribing a Given Parabola

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

If an equilateral triangle circumscribes a parabola, that is, its sides (extended if necessary) are tangent to the parabola, then its center moves along a straight line, which is none other than the parabola's directrix. To prove this was a question in the oral examination of the Ecole Polytechnique in 1928.

Contributed by: Emmanuel Amiot (March 2011)
Open content licensed under CC BY-NC-SA



A parabola is the locus of points equidistant from a given point (the focus) and a given line (the directrix). A proof could be obtained from projective geometry, where the tangents to the parabola themselves define a conic section in another space. But Mathematica computes the coordinates of the vertices and center of the triangle easily enough, starting from any tangent to the parabola. As a bonus, the computation yields that the vertices of this triangle are located on a hyperbola (see the blue dot on the red curve and observe the other vertices of the triangle).

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.