Circumscribed Circle - Triangles

Triangles

All triangles are cyclic, i.e. every triangle has a circumscribed circle.

The circumcenter of a triangle can be found as the intersection of any two 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 vertices, and all points on the perpendicular bisectors are equidistant from two of the vertices of the triangle.

Alternate method to determine the circumcenter: draw any two lines departing the vertices at an angle with the common side, equal to 90 degrees minus the angle of the opposite vertex.

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.

The circumcenter's position depends on the type of triangle:

• If and only if a triangle is acute (all angles smaller than a right angle), the circumcenter lies inside the triangle
• If and only if it is obtuse (has one angle bigger than a right angle), the circumcenter lies outside
• If and only if it is a right triangle, the circumcenter lies at the center of the hypotenuse. This is one form of Thales' theorem.

• The circumcenter of an acute triangle is inside the triangle

• The circumcenter of a right triangle is at the center of the hypotenuse

• The circumcenter of an obtuse triangle is outside the triangle

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 does not 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 diameter of the circumcircle of the triangle ΔABC is

where a, b, c are the lengths of the sides of the triangle and s = (a + b + c)/2 is the semiperimeter. The expression above is the area of the triangle, by Heron's formula. Trigometric expressions for the diameter of the circumcircle include

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.