plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

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

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. More specifically, consider a triangle , and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the triangle (these may need to be produced, i.e., extended). Label the intersections of the lines from with the sides . The pedal triangle is then . If is not an

obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a trian ...

and is the

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

, then the angles of are , and . The quadrilaterals are cyclic quadrilaterals. The location of the chosen point relative to the chosen triangle gives rise to some special cases: * If is the

, then is the orthic triangle. * If is 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 bis ...

, then is the

intouch 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. ...

. * If is the

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

, then is the

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 no ...

. *If is on the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

of the triangle, collapses to a line (the ''pedal line'' or ''

Simson line In geometry, given a triangle and a Point (geometry), 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 ...

'').

The vertices of the pedal triangle of an interior point , as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:

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

Trilinear coordinates

If 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 ...

, then the vertices of the pedal triangle of are given by

N &=& p+r\cos B &:& q+r\cos A &:& 0 \end

Antipedal triangle

One vertex, , of the antipedal triangle of is the point of intersection of the perpendicular to through and the perpendicular to through . Its other vertices, and , 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

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) \end

For example, the excentral triangle is the antipedal triangle of the incenter. Suppose that does not lie on any of the extended sides , and let 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 . The pedal triangle of is homothetic to the antipedal triangle of . The homothetic center (which is a triangle center if and only if is a triangle center) is the point given in

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 and the antipedal triangle of equals the square of the area of .

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

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

pedal triangle and pedal circle
- interactive illustration Objects defined for a triangle