Ceviana

Em geometria, as cevianas são segmentos de reta que partem do vértice do triângulo para o lado oposto. Medianas, alturas e bissetrizes são casos especiais de cevianas. O nome ceviana vem do engenheiro italiano Giovanni Ceva, que formulou o Teorema de Ceva, que dá condições para que três cevianas sejam concorrentes.

Comprimento da ceviana

Ver artigo principal: Teorema de Stewart

O comprimento de uma ceviana pode ser determinado pelo Teorema de Stewart. No diagrama, o comprimento $d$ pode ser determinado através da fórmula

\,b^{2}m+c^{2}n=a(d^{2}+mn).

Se a ceviana é uma mediana, o seu comprimento pode ser determinado através da fórmula

\,m(b^{2}+c^{2})=a(d^{2}+m^{2})

\,2(b^{2}+c^{2})=4d^{2}+a^{2}

já que

\,a=m+n.

Se a ceviana é uma bissetriz, o seu comprimento pode ser determinado através da fórmula

\,(b+c)^{2}=a^{2}\left({\frac {d^{2}}{mn}}+1\right).

Se a ceviana é uma altura, o seu comprimento pode ser determinado através da fórmula

\,d^{2}=b^{2}-n^{2}=c^{2}-m^{2}.

Ver também

Referências

Ross Honsberger (1995) Episodes in Nineteenth and Twentieth Century Euclidean Geometry, pages 13 and 137, Mathematical Association of America.
Vladimir Karapetoff (1929) "Some properties of correlative vertex lines in a plane triangle", American Mathematical Monthly 36:476–9.
Indika Shameera Amarasinghe (2011) A New Theorem on any Right-angled Cevian Triangle, Journal of the World Federation of National Mathematics Competitions, Vol 24(02), pp.29 - 37. (isso é a prova que wikipedia ñ é confiavel )

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