geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a pedal triangle is obtained by projecting a
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
onto the sides of a
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
.
More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the three sides of the triangle (these may need to be produced, i.e., extended). Label ''L'', ''M'', ''N'' the intersections of the lines from ''P'' with the sides ''BC'', ''AC'', ''AB''. The pedal triangle is then ''LMN''.
If ABC is not an obtuse triangle, P is the orthocenter then the angles of LMN are 180°−2A, 180°−2B and 180°−2C.
The location of the chosen point ''P'' relative to the chosen triangle ''ABC'' gives rise to some special cases:
* If ''P = ''
orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
, then ''LMN = ''intouch triangle.
* If ''P = ''
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, then ''LMN = ''
medial triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...
.
If ''P'' is on the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of the triangle, ''LMN'' collapses to a line. This is then called the pedal line, or sometimes the
Simson line
In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, however ...
after
Robert Simson
Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.Carnot's theorem:
:
Trilinear coordinates
If ''P'' has
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
''p'' : ''q'' : ''r'', then the vertices ''L,M,N'' of the pedal triangle of ''P'' are given by
*''L = 0 : q + p'' cos C'' : r + p ''cos'' B''
*''M = p + q ''cos'' C : 0 : r + q ''cos'' A''
*''N = p + r ''cos'' B : q + r ''cos'' A : 0''
Antipedal triangle
One vertex, ''L, of the antipedal triangle of ''P'' is the point of intersection of the perpendicular to ''BP'' through ''B'' and the perpendicular to ''CP'' through ''C''. Its other vertices, ''M'' ' and ''N'' ', are constructed analogously.
Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
are given by
*''L ''= − (q + p'' cos'' C)(r + p'' cos'' B) : (r + p'' cos'' B)(p + q'' cos'' C) : (q + p'' cos'' C)(p + r'' cos'' B)''
*''M ''= (r + q'' cos'' A)(q + p'' cos'' C) : − (r + q'' cos'' A)(p + q'' cos'' C) : (p + q'' cos'' C)(q + r'' cos'' A)''
*''N ''= (q + r'' cos'' A)(r + p'' cos'' B) : (p + r'' cos'' B)(r + q'' cos'' A) : − (p + r'' cos'' B)(q + r'' cos'' A)''
For example, the
excentral triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
is the antipedal triangle of the incenter.
Suppose that ''P'' does not lie on any of the extended sides ''BC, CA, AB,'' and let ''P''−1 denote the
isogonal conjugate __notoc__
In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
of ''P''. The pedal triangle of ''P'' is homothetic to the antipedal triangle of ''P''−1. The homothetic center (which is a triangle center if and only if ''P'' is a triangle center) is the point given in
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
by
: ''ap(p + q'' cos'' C)(p + r'' cos'' B) : bq(q + r'' cos'' A)(q + p'' cos'' C) : cr(r + p'' cos'' B)(r + q'' cos'' A)''.
The product of the areas of the pedal triangle of ''P'' and the antipedal triangle of ''P''−1 equals the square of the area of triangle ''ABC''.
Pedal circle
The pedal circle is defined as the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of the pedal triangle. Note that the pedal circle is not defined for points lying on the circumcircle of the triangle.
Pedal circle of isogonal conjugates
For any point not lying on the circumcircle of the triangle, it is known that and its isogonal conjugate have a common pedal circle, whose center is the midpoint of these two points.