## Nickalls' Theorem

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

This beautiful theorem crystalized from experimental observations in 1986 by Dick Nickalls (retired) from the Dept. of Anaesthesia, Nottingham University Hospitals, UK in the area of vision physiology - a rich source of mathematical problems. It relates to the so-called Pulfrich effect, an optical illusion in which an object which is moving in a plane parallel to the viewer's forehead seems to move out of that plane and to approach (or recede from) the viewer.

References

1. Nickalls, R.W.D. (2000). A conic theorem generalised: directed angles and applications. The Mathematical Gazette, 84 (July), 232–241. (Download).
2. Rigby, J.F (2002). A proof of Nickalls' theorem on tangents and foci. The Mathematical Gazette, 86 (July), 322-324. (Download).

Please WAIT while the applet loads. NOTE: If a security pop-up menu appears in your browser, please choose RUN/ALLOW to let the GeoGebra applet run properly. It is completely safe & can be trusted. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applet loading, please go here for additional information on Java settings that should resolve the issue.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Drag points A, B, C, D or E and observe the theorem dynamically. Also drag the ellipse until it becomes a circle. What do you notice?

Click on hyperbola or parabola to dynamically interact with and investigate these other two cases of Nickalls' theorem.

The 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