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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. Addendum to "The Provenance of Urquhart's Theorem." Austral. Math. Soc. Gaz. 9, 100, 1982.Deakin, M. A. B. "Yet More on Urquhart's Theorem." http://www.austms.org.au/Publ/Gazette/1997/Apr97/letters.html.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 Web Resource. https://mathworld.wolfram.com/UrquhartsTheorem.html

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