Circumcircle

In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. Every bounded shape has a unique circumcircle. The center of this circumcircle is known as the shape's circumcenter.

Contents

1 Cyclic polygons 2 Circumcircles of triangles 2.1 Circumcircle equation 3 Circumcircle of circles 4 See also 5 External links

Cyclic polygons

At least two vertices of a shape will lie on its circumcircle. A polygon whose vertices all lie on its circumcircle is said to be a cyclic polygon. All regular simple polygons, all triangles and all rectangles are cyclic.

Circumcircles of triangles

The circumcircle of a triangle is the unique circle on which all its three vertices lie. (This is not the same as the first definition for "thin" triangles where only two points would lie on the first definition's circumcircle.) 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.

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

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Circumcircles_of_triangles.png
Image:Circumcircles of triangles.png

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.

Circumcircle equation

The circumcircle is given by the equation

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

Given

<math>a=\det\begin{vmatrix}

A_x & A_y & 1 \\ B_x & B_y & 1 \\ C_x & C_y & 1 \end{vmatrix}<math>, <math> 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}<math>, <math> 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}<math>, <math> b=\det\begin{vmatrix} A_x & A_y & A^2 \\ B_x & B_y & B^2 \\ C_x & C_y & C^2 \end{vmatrix}<math> 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</sub> = 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.}

Circumcircle of circles

See Descartes' theorem.

See also

• incircle

External links

• Triangle centers (http://agutie.homestead.com/files/Trianglecenter.html) by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Transitivity in Action (http://www.cut-the-knot.org/triangle/remarkable.shtml) Remarkable Points in a Triangle
• Angle Trisectors on Circumcircle (http://www.cut-the-knot.org/Curriculum/Geometry/Roussel.shtml)
• Circumcevian Triangle (http://www.cut-the-knot.org/Curriculum/Geometry/OrthoCircum.shtml)
• Concyclic Circumcenters: A Dynamic View (http://www.cut-the-knot.org/ctk/SixCircum.shtml)
• Isogonal image of the circumcircle (http://www.cut-the-knot.org/Curriculum/Geometry/Circumsogonal.shtml)
• Reflections of a Point on the Circumcircle (http://www.cut-the-knot.org/ctk/ThroughOrthocenter.shtml)it:Circumcerchio

nl:Omgeschreven cirkel zh:外接圓