Reuschle's theorem

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.

Reuschle's theorem:
cevians ${\displaystyle AP_{a}}$, ${\displaystyle AP_{b}}$ and ${\displaystyle AP_{c}}$ intersect in ${\displaystyle D}$
${\displaystyle AP'_{a}}$, ${\displaystyle AP'_{b}}$ and ${\displaystyle AP'_{c}}$ intersect in ${\displaystyle D'}$

In a triangle ${\displaystyle ABC}$ with its three cevians intersecting in a common point other than the vertices ${\displaystyle A}$, ${\displaystyle B}$ or ${\displaystyle C}$ let ${\displaystyle P_{a}}$, ${\displaystyle P_{b}}$ and ${\displaystyle P_{c}}$ denote the intersections of the (extended) triangle sides and the cevians. The circle defined by the three points ${\displaystyle P_{a}}$, ${\displaystyle P_{b}}$ and ${\displaystyle P_{c}}$ intersects the (extended) triangle sides in the (additional) points ${\displaystyle P'_{a}}$, ${\displaystyle P'_{b}}$ and ${\displaystyle P'_{c}}$. Reuschle's theorem now states that the three new cevians ${\displaystyle AP'_{a}}$, ${\displaystyle BP'_{b}}$ and ${\displaystyle CP'_{c}}$ intersect in a common point as well.

References

• Friedrich Riecke (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.