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Urquhart's Theorem

UrquhartsTheorem

If ABB^' and AC^'C are straight lines with BC and B^'C^' intersecting at D and AB+BD=AC^'+C^'D, then AB^'+B^'D=AC+CD.

The origin and some history of this theorem are discussed by Pedoe (1976) who attributed it to L. M. Urquhart. However, de Morgan had published a proof of the theorem in 1841, and the theorem may be viewed as a limiting case of a result due to Chasles that dates back to 1860 (Deakin 1981, Deakin 1982, Hajja 2006).

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References

Deakin, M. A. B. "The Provenance of Urquhart's Theorem." Austral. Math. Soc. Gaz. 8, 26, 1981.Deakin, M. A. B. "Yet More on Urquhart's Theorem." http://www.austms.org.au/Publ/Gazette/1997/Apr97/letters.html.Deakin, M. A. B. Addendum to "The Provenance of Urquhart's Theorem." Austral. Math. Soc. Gaz. 9, 100, 1982.Eustice, D. "Urquhart's Theorem and the Ellipse." Crux Math. (Eureka), 2, 132-133, 1976.Grossman, H. "Urquhart's Quadrilateral Theorem." Math. Teacher 66, 643-644, 1973.Hajja, M. "An Elementary Proof of the Most 'Elementary' Theorem of Euclidean Geometry." J. Geom. Graphics 8, 17-22, 2004.Hajja, M. "A Very Short and Simple Proof of 'the Most Elementary Theorem' of Euclidean Geometry." Forum Geom. 6, 167-169, 2006.Kazarinoff, N. D. "Geometric Inequalities." Washington, DC: Math. Assoc. Amer., 1961.Konhauser, J. D. E.; Velleman, D.; and Wagon, S. Which Way Did the Bicycle Go? And Other Intriguing Mathematical Mysteries. Washington, DC: Math. Assoc. Amer., 1996.Pedoe, D. "The Most 'Elementary' Theorem of Euclidean Geometry." Math. Mag. 49, 40-42, 1976.Sauvé, L. "On Circumscribable Quadrilaterals." Crux Math. (Eureka), 2, 63-67, 1976.Sokolowsky, D. "Extensions of Two Theorems by Grossman." Crux Math. (Eureka) 2, 163-170, 1976.Sokolowsky, D. "A 'No-Circle' Proof of Urquhart's Theorem." Crux Math. (Eureka) 2, 133-134, 1976.Trost, E. and Breusch, R. Problem 4964. Amer. Math. Monthly 68, 384, 1961.Trost, E. and Breusch, R. Solution to Problem 4964. Amer. Math. Monthly 69, 672-674, 1962.Williams, K. S. "Pedoe's Formulation of Urquhart's Theorem." Ontario Math. Gaz. 15, 42-44, 1976.Williams, K. S. "On Urquhart's Elementary Theorem of Euclidean Geometry." Crux Math. (Eureka) 2, 108-109, 1976.

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Urquhart's Theorem

Cite this as:

Weisstein, Eric W. "Urquhart's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UrquhartsTheorem.html

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