Nickalls' Theorem (parabola case)

If tangents from two points E and D are drawn to a parabola with focus A and directrix BC, and these tangents respectively intersect at F and G as shown below, then ∠EFD + ∠EGD = ∠EAD [or as shown by the dynamic measurements on the bottom left α + β (= γ) = δ ].

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Drag points A, B, C, D or E and observe the theorem dynamically.

The parabola case of Nickall's theorem is also useful in providing easy proofs for other fundamental properties of conics such as "The circumcircle of the triangle formed by any three tangents to a parabola passes through the focus" (e.g. see Parabola Circumcircle Theorem), "The locus of the foot of the perpendicular from the focus of a parabola to a tangent is the tangent at the apex" (e.g. see Parabola Locus Theorem), etc.

Michael de Villiers, updated 16 March 2012, created with GeoGebra