Given a point 
,
the pedal triangle of 
is the triangle whose polygon
vertices are the feet of the perpendiculars from 
to the side lines. The pedal triangle of a triangle
with trilinear coordinates 
and angles 
, 
,
and 
has trilinear vertex matrix
![[0 beta+alphacosC gamma+alphacosB; alpha+betacosC 0 gamma+betacosA; alpha+gammacosB beta+gammacosA 0]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FPedalTriangle%2FNumberedEquation1.svg) |
(1) |
(Kimberling 1998, p. 186), and is a central triangle of type 2 (Kimberling 1998, p. 55).
The side lengths are
where 
is the circumradius of 
, and area is
 |
(5) |
where 
is the area of 
.
The following table summarizes a number of special pedal triangles for various special pedal points 
.
The symmedian point of a triangle is the triangle centroid of its pedal triangle (Honsberger 1995, pp. 72-74).
The third pedal triangle is similar to the original one. This theorem can be generalized to: the 
th
pedal 
-gon
of any 
-gon
is similar to the original one. It is also true that
 |
(6) |
(Johnson 1929, pp. 135-136; Stewart 1940; Coxeter and Greitzer 1967, p. 25). The area 
of the pedal triangle of a point 
is proportional to the power of
with respect to the circumcircle,
(Johnson 1929, pp. 139-141).
The only closed billiards path of a single circuit in an acute triangle is the pedal triangle. There are an infinite number of multiple-circuit paths, but all segments are parallel to the sides of the pedal triangle (Wells 1991).
See also
Antipedal Triangle, Fagnano's Problem, Orthic Triangle, Pedal Circle, Pedal Line
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References
Coxeter, H. S. M. and Greitzer, S. L. "Pedal Triangles." ยง1.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 22-26, 1967.Gallatly, W. "Pedal Triangles." Ch. 5 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 37-45, 1913.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 67-74, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Stewart, B. M. "Cyclic Properties of Miquel Polygons." Amer. Math. Monthly 47, 462-466, 1940.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.
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Cite this as:
Weisstein, Eric W. "Pedal Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PedalTriangle.html