Antipedal Triangle

The antipedal triangle $A$ of a given Triangle $T$ is the Triangle of which $T$ is the Pedal Triangle. For a Triangle with Trilinear Coordinates $\alpha:\beta:\gamma$ and Angles $A$, $B$, and $C$, the antipedal triangle has Vertices with Trilinear Coordinates

$$-(\beta+\alpha\cos C)(\gamma+\alpha\cos B):(\gamma+\alpha\co... ...(\alpha+\beta\cos C):(\beta+\alpha\cos C)(\alpha+\gamma\cos B)$$

$$(\gamma+\beta\cos A)(\beta+\alpha\cos C):-(\gamma+\beta\cos A)(\alpha+\beta\cos C):(\alpha+\beta\cos C)(\beta+\gamma\cos A)$$

$$(\beta+\gamma\cos A)(\gamma+\alpha\cos B):(\alpha+\gamma\cos... ...gamma+\beta\cos A):-(\alpha+\gamma\cos B)(\beta+\gamma\cos A).$$

The Isogonal Conjugate of the Antipedal Triangle of a given Triangle is Homothetic with the original Triangle. Furthermore, the Product of their Areas equals the Square of the Area of the original Triangle (Gallatly 1913).

See also Pedal Triangle

References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 56-58, 1913.