Reuschle's theorem

Reuschle's theorem Reuschle's theorem: cevians {displaystyle AP_{a}} , {displaystyle AP_{b}} e {displaystyle AP_{c}} intersect in {estilo de exibição D} {displumaystyle AP'_{a}} , {displaystyle AP'_{b}} e {displaystyle AP'_{c}} intersect in {displaystyle D'} 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 {estilo de exibição ABC} with its three cevians intersecting in a common point other than the vertices {estilo de exibição A} , {estilo de exibição B} ou {estilo de exibição C} deixar {estilo de exibição P_{uma}} , {estilo de exibição P_{b}} e {estilo de exibição P_{c}} denote the intersections of the (extended) triangle sides and the cevians. The circle defined by the three points {estilo de exibição P_{uma}} , {estilo de exibição P_{b}} e {estilo de exibição P_{c}} intersects the (extended) triangle sides in the (adicional) pontos {displumaystyle P'_{a}} , {displaystyle P'_{b}} e {displaystyle P'_{c}} . Reuschle's theorem now stumates that the three new cevians {displaystyle AP'_{a}} , {displaystyle BP'_{b}} e {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 (Alemão) M. D. Raposa, J. R. Goggins: ">

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