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Antipedal Triangle


AntipedalTriangle

The antipedal triangle DeltaA^'B^'C^' of a reference triangle DeltaABC with respect to a given point P is the triangle of which DeltaABC is the pedal triangle with respect to P. If the point P has trilinear coordinates alpha:beta:gamma and the angles of DeltaABC are A, B, and C, then the antipedal triangle has trilinear vertex matrix

 [-(beta+alphacosC)(gamma+alphacosB) (gamma+alphacosB)(alpha+betacosC) (beta+alphacosC)(alpha+gammacosB); (gamma+betacosA)(beta+alphacosC) -(gamma+betacosA)(alpha+betacosC) (alpha+betacosC)(beta+gammacosA); (beta+gammacosA)(gamma+alphacosB) (alpha+gammacosB)(gamma+betacosA) -(alpha+gammacosB)(beta+gammacosA)]
(1)

(Kimberling 1998, p. 187).

The antipedal triangle is a central triangle of type 2 (Kimberling 1998, p. 55).

The following table summarizes some named antipedal triangles with respect to special antipedal points.

The antipedal triangle with respect to DeltaABC and P=alpha:beta:gamma has side lengths

a^'=(2R|calphabeta+balphagamma+abetagamma|sqrt(beta^2+gamma^2+2betagammacosA))/(|betagamma(aalpha+bbeta+cgamma)|)
(2)
b^'=(2R|calphabeta+balphagamma+abetagamma|sqrt(alpha^2+gamma^2+2alphagammacosB))/(|alphagamma(aalpha+bbeta+cgamma)|)
(3)
c^'=(2R|calphabeta+balphagamma+abetagamma|sqrt(alpha^2+beta^2+2alphabetacosC))/(|alphabeta(aalpha+bbeta+cgamma)|),
(4)

where R is the circumradius of DeltaABC, and area

 Delta^'=((calphabeta+balphagamma+abetagamma)^2R)/(alphabetagamma(aalpha+bbeta+cgamma)).
(5)

The isogonal conjugate of the antipedal triangle of a given triangle DeltaABC with respect to a point P is the antipedal triangle of DeltaABC with respect to the isogonal conjugate of P. It is also homothetic with the pedal triangle of DeltaABC with respect to P. Furthermore, the product of the areas of the two homothetic triangles equals the square of the area of the original triangle (Gallatly 1913, pp. 56-58).


See also

Antipedal Line, Pedal Triangle

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References

Gallatly, W. "Antipedal Triangles." Ch. 7 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 55-62, 1913.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Antipedal Triangle

Cite this as:

Weisstein, Eric W. "Antipedal Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntipedalTriangle.html

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