Basic Examples (2) 

Compute the three contact parabolas given affine contact locations on each side of the reference triangle opposite to the vertices:

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Tabulate the results above:

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Scope (2) 

Visualize three external contact parabolas, the contact points and the reference triangle using ContourPlot:

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Visualize three external contact parabolas and the reference triangle using ParametricPlot with parametric expressions:

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Properties and Relations (3) 

Visualize three external contact parabolas and their foci (blue). The foci are on the circumcircle of triangle ABC according to Lambert's theorem:

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The lines (orange) that link contact points and opposite vertices are concurrent at the generalized Gergonne point:

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The area of the triangle formed by the contacts is always twice the area of the reference triangle:

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The directrix (orange) of the external contact parabola passes through the orthocenter H of the reference triangle ABC:

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Possible Issues (2) 

Exparabola does not evaluate for degenerate triangles:

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Exparabola does not evaluate if the affine location of the contact point is not strictly within the side of the triangle:

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Neat Examples (2) 

Plot the external contact parabolas by fixing the first affine parameter. According to Pascal's theorem, there are infinite number of set of {μ₁,μ₂} in the configuration below such that the six external contact points (on the extension line of each side) share the same conic section:

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Nine point cubic curves pass though six external tangent point and three foci:

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