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 ...

, a pedal triangle is obtained by projecting a point onto the sides of a

triangle A triangle is a polygon with three edges and three 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, any three points, when non- colline ...

. 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 '' ...

, then ''LMN = '' orthic triangle. * If ''P = ''

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 bis ...

, 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, howeve ...

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: :

AN^2+BL^2+CM^2=NB^2+LC^2+MA^2.

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 ...

''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 ...

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 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

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

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

P

not lying on the circumcircle of the triangle, it is known that

P

and its isogonal conjugate

P^\star

have a common pedal circle, whose center is the midpoint of these two points.

References

{{Reflist

External links

Mathworld: Pedal Triangle

Simson Line

Pedal Triangle and Isogonal Conjugacy
Objects defined for a triangle