Triángulo extratangente

En geometría, el triángulo extratangente de un triángulo se forma al unir los puntos en los que las tres circunferencias exinscritas son tangentes al triángulo.

Coordenadas

Los vértices del triángulo extratangentes se dan en coordenadas trilineales por:

T_{A}=0:\csc ^{2}{\left(B/2\right)}:\csc ^{2}{\left(C/2\right)}

T_{B}=\csc ^{2}{\left(A/2\right)}:0:\csc ^{2}{\left(C/2\right)}

T_{C}=\csc ^{2}{\left(A/2\right)}:\csc ^{2}{\left(B/2\right)}:0

o equivalentemente, siendo a, b, c las longitudes de los lados opuestos a los ángulos A, B, C respectivamente,

T_{A}=0:{\frac {a-b+c}{b}}:{\frac {a+b-c}{c}}

T_{B}={\frac {-a+b+c}{a}}:0:{\frac {a+b-c}{c}}

T_{C}={\frac {-a+b+c}{a}}:{\frac {a-b+c}{b}}:0.

Figuras relacionadas

Las divisorias del triángulo son rectas que conectan los vértices del triángulo original con los vértices correspondientes del triángulo extratangente; que bisecan el perímetro del triángulo y se encuentran en el punto de Nagel. Este punto figura en color azul y está marcado como "N" en el diagrama.

La inelipse de Mandart es tangente a los lados del triángulo de referencia en los tres vértices del triángulo extratangente.^[1]

Área

El área del triángulo extratangente, $K_{T}$ , viene dada por:

K_{T}=K{\frac {2r^{2}s}{abc}}

donde $K$ , $r$ y $s$ son el área, el radio de la circunferencia exinscrita y el semiperímetro del triángulo original, y $a$ , $b$ , y $c$ son las longitudes de los lados del triángulo original.

Es la misma área que la del triángulo de las tangentes del incentro.^[2]

Referencias

↑ Juhász, Imre (2012), «Control point based representation of inellipses of triangles», Annales Mathematicae et Informaticae 40: 37-46, MR 3005114 ..
↑ Weisstein, Eric W. "Extouch Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExtouchTriangle.html

Datos: Q5422106

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