Circumscribed circle

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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which contains all the vertices of the polygon. The centre of this circle is called the circumcenter.

A polygon which has a circumscribed circle is called a cyclic polygon. All regular simple polygons, all triangles and all rectangles are cyclic.

A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. Not every polygon has a circumscribed circle, as the vertices of a polygon do not need to all lie on a circle. Yet any polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm ^[1]. Even if a polygon has a circumscribed circle, it may not coincide with its minimum bounding circle; for example, for an obtuse triangle, the minimum bounding circle has the hypotenuse as diameter and does not pass through the opposite vertex.

1 Circumcircles of triangles
- 1.1 Circumcircle equation
- 1.2 Coordinates of circumcenter
2 See also
3 References
4 External links

[edit] Circumcircles of triangles

The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular bisectors are equidistant from those points of 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.

A triangle is acute (all angles smaller than a right angle) if and only if the circumcenter lies inside the triangle; it is obtuse (has an angle bigger than a right one) if and only if the circumcenter lies outside, and it is a right triangle if and only if the circumcenter lies on one of its sides (namely on the hypotenuse). This is one form of Thales' theorem.

The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) The triangle's nine-point circle has half the diameter of the circumcircle.

The circumcenter always lies on one line with the triangle's centroid and orthocenter. This line is known as Euler's line.

The circumcenter's isogonal conjugate 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 an infinitely large circle. Nearly collinear points often cause frequent problems and errors in computation of the circumcircle.

Circumcircles of triangles have an intimate relationship with the Delaunay triangularization of a set of points.

[edit] Circumcircle equation

The circumcircle is given by the equation

$\det\begin{vmatrix} v^2 & v_x & v_y & 1 \\ A^2 & A_x & A_y & 1 \\ B^2 & B_x & B_y & 1 \\ C^2 & C_x & C_y & 1 \end{vmatrix}=0$

where A, B and C are the vertices of the triangle, and the solution for v is the circumcircle. (Note A² = A_x² + A_y².)

Given

$a=\det\begin{vmatrix} A_x & A_y & 1 \\ B_x & B_y & 1 \\ C_x & C_y & 1 \end{vmatrix}$ , $S_x=\frac{1}{2}\det\begin{vmatrix} A^2 & A_y & 1 \\ B^2 & B_y & 1 \\ C^2 & C_y & 1 \end{vmatrix}$ , $S_y=\frac{1}{2}\det\begin{vmatrix} A_x & A^2 & 1 \\ B_x & B^2 & 1 \\ C_x & C^2 & 1 \end{vmatrix}$ , $b=\det\begin{vmatrix} A_x & A_y & A^2 \\ B_x & B_y & B^2 \\ C_x & C_y & C^2 \end{vmatrix}$

we then have av² − 2Sv − b = 0, and assuming the three points were not in a line (otherwise the circumcircle doesn't exist), (v − S)² = b/a + S²/a², giving the circumcenter S/a and the circumradius √(b/a + S²/a²). This approach should also work for the circumsphere of a tetrahedron.

[edit] Coordinates of circumcenter

The circumcenter has trilinear coordinates (cos $α$ , cos $β$ , cos $γ$ ) where $α,β,γ$ are the angles of the triangle. The circumcenter has barycentric coordinates

$\left(a^2(-a^2+b^2+c^2),\;b^2(a^2-b^2+c^2),\;c^2(a^2+b^2-c^2)\right),$