Reuschle's theorem
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Reuschle's theorem:
cevians , and intersect in
, and intersect in
In elementary geometry, Reuschle's theorem describes a property of the cevians of a triangle intersecting in a common point and is named after the German mathematician Karl Gustav Reuschle (1812–1875). It is also known as Terquem's theorem after the French mathematician Olry Terquem (1782–1862), who published it in 1842.
In a triangle with its three cevians intersecting in a common point other than the vertices , or let , and denote the intersections of the (extended) triangle sides and the cevians. The circle defined by the three points , and intersects the (extended) triangle sides in the (additional) points , and . Reuschle's theorem now states that the three new cevians , and intersect in a common point as well.
References[]
- (ed.): Mathematische Unterhaltungen. Volume I, Stuttgart 1867, (reprint Wiesbaden 1973), ISBN 3-500-26010-1, p. 125 (German)
- M. D. Fox, J. R. Goggins: "Cevian Axes and Related Curves." The Mathematical Gazette, volume 91, no. 520, 2007, pp. 3-4 (JSTOR).
External links[]
- Terquem's theorem at cut-the-knot.org
- Weisstein, Eric W. "Cyclocevian Conjugate". MathWorld.
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Categories:
- Elementary geometry
- Theorems about triangles and circles
- Elementary geometry stubs