Concyclic points

In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle.

Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle.

In general the centre O of a circle on which points P and Q lie must be such that OP and OQ are equal distances.

Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter are concyclic.

[6] A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s = (a + b + c + d) / 2 has its circumradius given by[7][8] an expression that was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.

By Ptolemy's theorem, if a quadrilateral is given by the pairwise distances between its four vertices A, B, C, and D in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides: If two lines, one containing segment AC and the other containing segment BD, intersect at X, then the four points A, B, C, D are concyclic if and only if[9] The intersection X may be internal or external to the circle.

A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle.

In all known cases, its diagonals also have rational lengths, though whether this is true for all possible Robbins pentagons is an unsolved problem.

Then for any point M on the minor arc A1An, the distances from M to the vertices satisfy[14] For a regular n-gon, if

are the distances from any point M on the circumcircle to the vertices Ai, then [15] Any regular polygon is cyclic.

Let θ1 be the angle spanned by one side of the cyclic polygon as viewed from the center of the circumscribing circle.

Every Heronian triangle and every Brahmagupta quadrilateral has a rational value for the tangent of the quarter angle, tan θk/4, for every value of k. Every known Robbins pentagon (has diagonals that have rational length and) has this property, though it is an unsolved problem whether every possible Robbins pentagon has this property.

This reverse relationship gives a way to generate cyclic polygons with integer area, sides, and diagonals.

Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as sk = 4qk / (1 + qk2).

Every set of points in the plane has a unique minimum bounding circle, which may be constructed by a linear time algorithm.

For example, for an obtuse triangle, the minimum bounding circle has the longest side as diameter and does not pass through the opposite vertex.