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Center of an Equilateral Triangle Circumscribing a Given Parabola

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.

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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).

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