Pedal Triangle

Given a point $P$, the pedal triangle of $P$ is the Triangle whose Vertices are the feet of the perpendiculars from $P$ to the side lines. The pedal triangle of a Triangle with Trilinear Coordinates $\alpha:\beta:\gamma$ and angles $A$, $B$, and $C$ has Vertices with Trilinear Coordinates

$0:\beta+\alpha\cos C:\gamma+\alpha\cos B$	(1)
$\alpha+\beta\cos C:0:\gamma+\beta\cos A$	(2)
$\alpha+\gamma\cos B:\beta+\gamma\cos A:0.$	(3)

The third pedal triangle is similar to the original one. This theorem can be generalized to: the $n$th pedal $n$-gon of any $n$-gon is similar to the original one. It is also true that

$$P_2P_3=A_1P\sin\alpha_1$$ (4)

(Johnson 1929, pp. 135-136). The Area $A$ of the pedal triangle of a point $P$ is proportional to the Power of $P$ with respect to the Circumcircle,

$$A={\textstyle{1\over 2}}(R^2-\overline{OP}^2)\sin\alpha_1\sin\alpha_2\sin\alpha_3={R^2-\overline{OP}^2\over 4R^2}\Delta$$ (5)

(Johnson 1929, pp. 139-141).

See also Antipedal Triangle, Fagnano's Problem, Pedal Circle, Pedal Line, Schwarz's Triangle Problem

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 22-26, 1967.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.