how to find the circumcenter of a triangle

Circumcenter

The circumcenter is the center of a triangle's circumcircle. It can be found as the intersection of the perpendicular bisectors. The trilinear coordinates of the circumcenter are

(1)

and the exact trilinear coordinates are therefore

(2)

where is the circumradius, or equivalently

(3)

The circumcenter is Kimberling center .

The distance between the incenter and circumcenter is , where is the circumradius and is the inradius.

Distances to a number of other named triangle centers are given by

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

where is the triangle triangle centroid, is the orthocenter, is the incenter, is the symmedian point, is the nine-point center, is the Nagel point, is the de Longchamps point, is the circumradius, is Conway triangle notation, and is the triangle area.

If the triangle is acute, the circumcenter is in the interior of the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse.

For an acute triangle,

(13)

where is the midpoint of side , is the circumradius, and is the inradius (Johnson 1929, p. 190).

Given an interior point, the distances to the polygon vertices are equal iff this point is the circumcenter. The circumcenter lies on the Brocard axis.

The following table summarizes the circumcenters for named triangles that are Kimberling centers.

triangle	Kimberling	circumcenter
anticomplementary triangle		orthocenter
circumcircle mid-arc triangle		circumcenter
circum-medial triangle		circumcenter
circumnormal triangle		circumcenter
circum-orthic triangle		circumcenter
circumtangential triangle		circumcenter
contact triangle		incenter
D-triangle		midpoint of and
Euler triangle		nine-point center
excentral triangle		Bevan point
extouch triangle		circumcenter of extouch triangle
Feuerbach triangle		nine-point center
first Brocard triangle		midpoint of Brocard diameter
first Morley triangle		first Morley center
first Yff circles triangle		internal similitude center of the circumcircle and incircle
Fuhrmann triangle		Fuhrmann center
hexyl triangle		incenter
inner Napoleon triangle		triangle centroid
inner Vecten triangle		complement of
Lucas tangents triangle		isogonal conjugate of
medial triangle		nine-point center
mid-arc triangle		incenter
orthic triangle		nine-point center
outer Napoleon triangle		triangle centroid
outer Vecten triangle		complement of
reference triangle		circumcenter
reflection triangle		-Ceva conjugate of
second Brocard triangle		midpoint of Brocard diameter
second Yff circles triangle		external similitude center of the circumcircle and incircle
Stammler triangle		circumcenter
tangential triangle		circumcenter of the tangential triangle

CircumcenterOrthocenter

The circumcenter and orthocenter are isogonal conjugates.

CircumOrthPedal

The orthocenter of the pedal triangle formed by the circumcenter concurs with the circumcenter itself, as illustrated above.

The circumcenter also lies on the Brocard axis and Euler line. It is the center of the circumcircle, second Brocard circle, and second Droz-Farny circle and lies on the Brocard circle and Lester circle. It also lies on the Jerabek hyperbola and the Darboux cubic, M'Cay cubic, Neuberg cubic, orthocubic, and Thomson cubic.

The complement of the circumcenter is the nine-point center.

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