How to find a parabola’s vertex from a chord and the tangents at its ends

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Given a parabola with vertex A and focus B and a chord CD let E be the intersection of the tangents at C and D. We create the line perpendicular to the AB through E. Let F and G be the intersections between this line and the lines DA extended and CA extended. EF is perpendicular to CF and DG.

We can combine this result with the observation that the axis of the parabola is parallel to the median of the chord/tangent/tangent triangle to give a construction for the vertex and focus of the parabola given the triangle.

Expressions

Name Input
Angle DGF Eq Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image
Angle CFE Eq Derive Input Maple Input MathML Input Mathematica Input Maxima Input Mupad Input TI-Nspire Input text Input Image