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e from it to any of the three vertices alternative construction edit alternative construction of the circumcenter intersection of broken lines an alternative method to determine the circumcenter is to draw any two lines each one departing from one of the vertices at an angle with the common side the common angle of departure being 90 minus the angle of the opposite vertex in the case of the opposite angle being obtuse drawing a line at a negative angle means going outside the triangle in coastal navigation a triangle s circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available the horizontal angle between two landmarks defines the circumcircle upon which the observer lies circumcircle equations edit cartesian coordinates edit in the euclidean plane it is possible to give explicitly an equation of the circumcircle in terms of the cartesian coordinates of the vertices of the inscribed triangle suppose that a a x a y b b x b y c c x c y displaystyle begin aligned mathbf a a_ x a_ y mathbf b b_ x b_ y mathbf c c_ x c_ y end aligned are the coordinates of points a b and c the circumcircle is then the locus of points v v x v y in the cartesian plane satisfying the equations v u 2 r 2 a u 2 r 2 b u 2 r 2 c u 2 r 2 displaystyle begin aligned mathbf v mathbf u 2 r 2 mathbf a mathbf u 2 r 2 mathbf b mathbf u 2 r 2 mathbf c mathbf u 2 r 2 end aligned guaranteeing that the points a b c and v are all the same distance r from the common center u of the circle using the polarization identity these equations reduce to the condition that the matrix v 2 2 v x 2 v y 1 a 2 2 a x 2 a y 1 b 2 2 b x 2 b y 1 c 2 2 c x 2 c y 1 displaystyle begin bmatrix mathbf v 2 2v_ x 2v_ y 1 mathbf a 2 2a_ x 2a_ y 1 mathbf b 2 2b_ x 2b_ y 1 mathbf c 2 2c_ x 2c_ y 1 end bmatrix has a nonzero kernel thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix det v 2 v x v y 1 a 2 a x a y 1 b 2 b x b y 1 c 2 c x c y 1 0 displaystyle det begin bmatrix mathbf v 2 v_ x v_ y 1 mathbf a 2 a_ x a_ y 1 mathbf b 2 b_ x b_ y 1 mathbf c 2 c_ x c_ y 1 end bmatrix 0 using cofactor expansion let s x 1 2 det a 2 a y 1 b 2 b y 1 c 2 c y 1 s y 1 2 det a x a 2 1 b x b 2 1 c x c 2 1 a det a x a y 1 b x b y 1 c x c y 1 b det a x a y a 2 b x b y b 2 c x c y c 2 displaystyle begin aligned s_ x frac 1 2 det begin bmatrix mathbf a 2 a_ y 1 mathbf b 2 b_ y 1 mathbf c 2 c_ y 1 end bmatrix 5pt s_ y frac 1 2 det begin bmatrix a_ x mathbf a 2 1 b_ x mathbf b 2 1 c_ x mathbf c 2 1 end bmatrix 5pt a det begin bmatrix a_ x a_ y 1 b_ x b_ y 1 c_ x c_ y 1 end bmatrix 5pt b det begin bmatrix a_ x a_ y mathbf a 2 b_ x b_ y mathbf b 2 c_ x c_ y mathbf c 2 end bmatrix end aligned we then have a v 2 2 sv b 0 where s s x s y and assuming the three points were not in a line otherwise the circumcircle is that line that can also be seen as a generalized circle with s at infinity v s a 2 b a s 2 a 2 giving the circumcenter s a and the circumradius b a s 2 a 2 a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron parametric equation edit a unit vector perpendicular to the plane containing the circle is given by n p 2 p 1 p 3 p 1 p 2 p 1 p 3 p 1 displaystyle widehat n frac p_ 2 p_ 1 times p_ 3 p_ 1 p_ 2 p_ 1 times p_ 3 p_ 1 hence given the radius r center p c a point on the circle p 0 and a unit normal of the plane containing the circle n textstyle widehat n one parametric equation of the circle starting from the point p 0 and proceeding in a positively oriented i e right handed sense about n displaystyle scriptstyle widehat n is the following r s p c cos s r p 0 p c sin s r n p 0 p c displaystyle mathrm r s mathrm p_ c cos left frac mathrm s mathrm r right p_ 0 p_ c sin left frac mathrm s mathrm r right left widehat n times p_ 0 p_ c right trilinear and barycentric coordinates edit an equation for the circumcircle in trilinear coordinates x y z is 2 a x b y c z 0 an equation for the circumcircle in barycentric coordinates x y z is a 2 x b 2 y c 2 z 0 the isogonal conjugate of the circumcircle is the line at infinity given in trilinear coordinates by ax by cz 0 and in barycentric coordinates by x y z 0 higher dimensions edit additionally the circumcircle of a triangle embedded in d dimensions can be found using a generalized method let a b and c be d dimensional points which form the vertices of a triangle we start by transposing the system to place c at the origin a a c b b c displaystyle begin aligned mathbf a mathbf a mathbf c mathbf b mathbf b mathbf c end aligned the circumradius r is then r a b a b 2 a b a b 2 sin θ a b 2 sin θ displaystyle r frac left mathbf a right left mathbf b right left mathbf a mathbf b right 2 left mathbf a times mathbf b right frac left mathbf a mathbf b right 2 sin theta frac left mathbf a mathbf b right 2 sin theta where θ is the interior angle between a and b the circumcenter p 0 is given by p 0 a 2 b b 2 a a b 2 a b 2 c displaystyle p_ 0 frac left mathbf a right 2 mathbf b left mathbf b right 2 mathbf a times mathbf a times mathbf b 2 left mathbf a times mathbf b right 2 mathbf c this formula only works in three dimensions as the cross product is not defined in other dimensions but it can be generalized to the other dimensions by replacing the cross products with following identities a b c a c b b c a a b c a c b a b c a b 2 a 2 b 2 a b 2 displaystyle begin aligned mathbf a times mathbf b times mathbf c mathbf a cdot mathbf c mathbf b mathbf b cdot mathbf c mathbf a mathbf a times mathbf b times mathbf c mathbf a cdot mathbf c mathbf b mathbf a cdot mathbf b mathbf c left mathbf a times mathbf b right 2 left mathbf a right 2 left mathbf b right 2 mathbf a cdot mathbf b 2 end aligned circumcenter coordinates edit cartesian coordinates edit the cartesian coordinates of the circumcenter u u x u y displaystyle u left u_ x u_ y right are u x 1 d a x 2 a y 2 b y c y b x 2 b y 2 c y a y c x 2 c y 2 a y b y u y 1 d a x 2 a y 2 c x b x b x 2 b y 2 a x c x c x 2 c y 2 b x a x displaystyle begin aligned u_ x frac 1 d left a_ x 2 a_ y 2 b_ y c_ y b_ x 2 b_ y 2 c_ y a_ y c_ x 2 c_ y 2 a_ y b_ y right 5pt u_ y frac 1 d left a_ x 2 a_ y 2 c_ x b_ x b_ x 2 b_ y 2 a_ x c_ x c_ x 2 c_ y 2 b_ x a_ x right end aligned with d 2 a x b y c y b x c y a y c x a y b y displaystyle d 2 left a_ x b_ y c_ y b_ x c_ y a_ y c_ x a_ y b_ y right without loss of generality this can be expressed in a simplified form after translation of the vertex a to the origin of the cartesian coordinate systems i e when a a a a x a y 0 0 in this case the coordinates of the vertices b b a and c c a represent the vectors from vertex a to these vertices observe that this trivial translation is possible for all triangles and the circumcenter u u x u y displaystyle u u _ x u _ y of the triangle a b c follow as u x 1 d c y b x 2 b y 2 b y c x 2 c y 2 u y 1 d b x c x 2 c y 2 c x b x 2 b y 2 displaystyle begin aligned u _ x frac 1 d left c _ y b _ x 2 b _ y 2 b _ y c _ x 2 c _ y 2 right 5pt u _ y frac 1 d left b _ x c _ x 2 c _ y 2 c _ x b _ x 2 b _ y 2 right end aligned with d 2 b x c y b y c x displaystyle d 2 b _ x c _ y b _ y c _ x due to the translation of vertex a to the origin the circumradius r can be computed as r u u x 2 u y 2 displaystyle r u sqrt u _ x 2 u _ y 2 and the actual circumcenter of abc follows as u u a displaystyle u u a trilinear coordinates edit the circumcenter has trilinear coordinates 3 cos α cos β cos γ where α β γ are the angles of the triangle in terms of the side lengths a b c the trilinears are 4 a b 2 c 2 a 2 b c 2 a 2 b 2 c a 2 b 2 c 2 displaystyle a left b 2 c 2 a 2 right b left c 2 a 2 b 2 right c left a 2 b 2 c 2 right barycentric coordinates edit the circumcenter has barycentric coordinates a 2 b 2 c 2 a 2 b 2 c 2 a 2 b 2 c 2 a 2 b 2 c 2 displaystyle a 2 left b 2 c 2 a 2 right b 2 left c 2 a 2 b 2 right c 2 left a 2 b 2 c 2 right 5 where a b c are edge lengths bc ca ab respectively of the triangle in terms of the triangle s angles α β γ displaystyle alpha beta gamma the barycentric coordinates of the circumcenter are 4 sin 2 α sin 2 β sin 2 γ displaystyle sin 2 alpha sin 2 beta sin 2 gamma circumcenter vector edit since the cartesian coordinates of any point are a weighted average of those of the vertices with the weights being the point s barycentric coordinates normalized to sum to unity the circumcenter vector can be written as u a 2 b 2 c 2 a 2 a b 2 c 2 a 2 b 2 b c 2 a 2 b 2 c 2 c a 2 b 2 c 2 a 2 b 2 c 2 a 2 b 2 c 2 a 2 b 2 c 2 displaystyle u frac a 2 left b 2 c 2 a 2 right a b 2 left c 2 a 2 b 2 right b c 2 left a 2 b 2 c 2 right c a 2 left b 2 c 2 a 2 right b 2 left c 2 a 2 b 2 right c 2 left a 2 b 2 c 2 right here u is the vector of the circumcenter and a b c are the vertex vectors the divisor here equals 16 s 2 where s is the area of the triangle as stated previously a a c b b c displaystyle begin aligned mathbf a mathbf a mathbf c mathbf b mathbf b mathbf c end aligned cartesian coordinates from cross and dot products edit in euclidean space there is a unique circle passing through any given three non collinear points p 1 p 2 and p 3 using cartesian coordinates to represent these points as spatial vectors it is possible to use the dot product and cross product to calculate the radius and center of the circle let p 1 x 1 y 1 z 1 p 2 x 2 y 2 z 2 p 3 x 3 y 3 z 3 displaystyle mathrm p_ 1 begin bmatrix x_ 1 y_ 1 z_ 1 end bmatrix mathrm p_ 2 begin bmatrix x_ 2 y_ 2 z_ 2 end bmatrix mathrm p_ 3 begin bmatrix x_ 3 y_ 3 z_ 3 end bmatrix then the radius of the circle is given by r p 1 p 2 p 2 p 3 p 3 p 1 2 p 1 p 2 p 2 p 3 displaystyle mathrm r frac left p_ 1 p_ 2 right left p_ 2 p_ 3 right left p_ 3 p_ 1 right 2 left left p_ 1 p_ 2 right times left p_ 2 p_ 3 right right the center of the circle is given by the linear combination p c α p 1 β p 2 γ p 3 displaystyle mathrm p_ c alpha p_ 1 beta p_ 2 gamma p_ 3 where α p 2 p 3 2 p 1 p 2 p 1 p 3 2 p 1 p 2 p 2 p 3 2 β p 1 p 3 2 p 2 p 1 p 2 p 3 2 p 1 p 2 p 2 p 3 2 γ p 1 p 2 2 p 3 p 1 p 3 p 2 2 p 1 p 2 p 2 p 3 2 displaystyle begin aligned alpha frac left p_ 2 p_ 3 right 2 left p_ 1 p_ 2 right cdot left p_ 1 p_ 3 right 2 left left p_ 1 p_ 2 right times left p_ 2 p_ 3 right right 2 beta frac left p_ 1 p_ 3 right 2 left p_ 2 p_ 1 right cdot left p_ 2 p_ 3 right 2 left left p_ 1 p_ 2 right times left p_ 2 p_ 3 right right 2 gamma frac left p_ 1 p_ 2 right 2 left p_ 3 p_ 1 right cdot left p_ 3 p_ 2 right 2 left left p_ 1 p_ 2 right times left p_ 2 p_ 3 right right 2 end aligned location relative to the triangle edit the circumcenter s position depends on the type of triangle for an acute triangle all angles smaller than a right angle the circumcenter always lies inside the triangle for a right triangle the circumcenter always lies at the midpoint of the hypotenuse this is one form of thales theorem for an obtuse triangle a triangle with one angle bigger than a right angle the circumcenter always lies outside the triangle the circumcenter of an acute triangle is inside the triangle the circumcenter of a right triangle is at the midpoint of the hypotenuse the circumcenter of an obtuse triangle is outside the triangle these locational features can be seen by considering the trilinear or barycentric coordinates given above for the circumcenter all three coordinates are positive for any interior point at least one coordinate is negative for any exterior point and one coordinate is zero and two are positive for a non vertex point on a side of the triangle angles edit the angles which the circumscribed circle forms with the sides of the triangle coincide with angles at which sides meet each other the side opposite angle α meets the circle twice once at each end in each case at angle α similarly for the other two angles this is due to the alternate segment theorem which states that the angle between the tangent and chord equals the angle in the alternate segment triangle centers on the circumcircle of triangle abc edit in this section the vertex angles are labeled a b c and all coordinates are trilinear coordinates steiner point bc b 2 c 2 ca c 2 a 2 ab a 2 b 2 the nonvertex point of intersection of the circumcircle with the steiner ellipse the steiner ellipse with center centroid abc is the ellipse of least area that passes through a b and c an equation for this ellipse is 1 ax 1 by 1 cz 0 tarry point sec a ω sec b ω sec c ω antipode of the steiner point focus of the kiepert parabola csc b c csc c a csc a b other properties edit the diameter of the circumcircle called the circumdiameter and equal to twice the circumradius can be computed as the length of any side of the triangle divided by the sine of the opposite angle diameter a sin a b sin b c sin c displaystyle text diameter frac a sin a frac b sin b frac c sin c as a consequence of the law of sines it does not matter which side and opposite angle are taken the result will be the same the diameter of the circumcircle can also be expressed as diameter a b c 2 area a b b c c a 2 δ a b c a b c 2 s s a s b s c 2 a b c a b c a b c a b c a b c displaystyle begin aligned text diameter frac abc 2 cdot text area frac ab bc ca 2 delta abc 5pt frac abc 2 sqrt s s a s b s c 5pt frac 2abc sqrt a b c a b c a b c a b c end aligned where a b c are the lengths of the sides of the triangle and s a b c 2 is the semiperimeter the expression s s a s b s c displaystyle sqrt scriptstyle s s a s b s c above is the area of the triangle by heron s formula 6 trigonometric expressions for the diameter of the circumcircle include 7 diameter 2 area sin a sin b sin c displaystyle text diameter sqrt frac 2 cdot text area sin a sin b sin c the triangle s nine point circle has half the diameter of the circumcircle in any given triangle the circumcenter is always collinear with the centroid and orthocenter the line that passes through all of them is known as the euler line the isogonal conjugate of the circumcenter is the orthocenter the useful minimum bounding circle of three points is defined either by the circumcircle where three points are on the minimum bounding circle or by the two points of the longest side of the triangle where the two points define a diameter of the circle it is common to confuse the minimum bounding circle with the circumcircle the circumcircle of three collinear points is the line on which the three points lie often referred to as a circle of infinite radius nearly collinear points often lead to numerical instability in computation of the circumcircle circumcircles of triangles have an intimate relationship with the delaunay triangulation of a set of points by euler s theorem in geometry the distance between the circumcenter o and the incenter i is o i r r 2 r displaystyle oi sqrt r r 2r where r is the incircle radius and r is the circumcircle radius hence the circumradius is at least twice the inradius euler s triangle inequality with equality only in the equilat...