Circumcircle

Encyclopedia : C : CI : CIR : Circumcircle

In geometry, the circumcircle is a unique circle associated with every two-dimensional geometric shape. There are two different definitions for the circumcircle depending on whether the shape is a triangle or not.

;Non-Triangles:The smallest circle that completely contains the shape within it. Every circumcircle of a polygon touches at least three vertices.
;Triangles:Due to the nature of the circumcircle touching at least three vertices, the circumcircle of a triangle is defined to touch the three vertices even though, for "thin circles", the smallest circle that exists only touches two vertices.

Therefore, the circumcircle is the smallest circle that contains a shape with the exception that, for triangles, it is a synonym for circumscribed circle.

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 three 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. The circumcircle and circumscribed circle of a cyclic polygon are identical.

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.

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.

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

[\det\beginv^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=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\beginA_x & A_y & 1 \\B_x & B_y & 1 \\C_x & C_y & 1\end], [S_x=\frac\det\beginA^2 & A_y & 1 \\B^2 & B_y & 1 \\C^2 & C_y & 1\end], [S_y=\frac\det\beginA_x & A^2 & 1 \\B_x & B^2 & 1 \\C_x & C^2 & 1\end], [b=\det\beginA_x & A_y & A^2 \\B_x & B_y & B^2 \\C_x & C_y & C^2\end]

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.

Circumcircle of circles

See: Descartes' theorem

See also

Incircle
External links

[Triangle centers] by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
[Transitivity in Action] at cut-the-knot
[Angle Trisectors on Circumcircle] at cut-the-knot
[Circumcevian Triangle] at cut-the-knot
[Concyclic Circumcenters: A Dynamic View] at cut-the-knot
[Isogonal image of the circumcircle] at cut-the-knot
[Reflections of a Point on the Circumcircle] at cut-the-knot
[Definition of circumcircle. Math Open Reference] With interactive applet

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.}