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Complement

In general, the word "complement" refers to that subset of some set which excludes a given subset . Taking and its complement together then gives the whole of the original set. The notations and are commonly used to denote the complement of a set .

This concept is commonly used and made precise in the particular cases of a complement point, graph complement, knot complement, and complement set. The word "complementary" is also used in the same way, so combining an angle and its complementary angle gives a right angle and a complementary error function erfc and the usual error function erf give unity when added together,

(1)

The complement point of a point with respect to a reference triangle , also called the inferior point, subordinate point, or medial image, is the point such that

(2)

where is the triangle centroid.

The complement point of a point with trilinear coordinates is therefore given by

(3)

The following table lists the complements of some named circles.

circle	complement
circumcircle	nine-point circle
anticomplementary circle	circumcircle
polar circle	de Longchamps circle

The complement of a line

(4)

is given by the line

(5)

The following table summarizes the complements of a number of named lines.

line	Kimberling	complement line
antiorthic axis
Brocard axis	*
de Longchamps line		orthic axis
Euler line		Euler line
Fermat axis	*
Gergonne line	*
Lemoine axis
line at infinity		line at infinity
Nagel line		Nagel line
orthic axis	*
Soddy line

The following table summarizes the complements of several common triangle centers.

	point		complement point
	incenter		Spieker center
	triangle centroid		triangle centroid
	circumcenter		nine-point center
	orthocenter		circumcenter
	nine-point center		midpoint of and
	symmedian point
	Gergonne point		mittenpunkt
	Nagel point		incenter
	mittenpunkt
	Spieker center
	first Fermat point
	second Fermat point
	first isodynamic point
	first Napoleon point
	second Napoleon point
	de Longchamps point		orthocenter
	Schiffler point
	Exeter point
	far-out point


	third power point

	Bevan point		midpoint of

	Kosnita point

	isogonal conjugate of
	isogonal conjugate of
	isogonal conjugate of

	isogonal conjugate of
	Prasolov point
	isotomic conjugate of orthocenter		symmedian point





	triangle vertex		midpoint of side
	triangle vertex		midpoint of side
	triangle vertex		midpoint of side
	equal parallelians point		isotomic conjugate of incenter