concyclic

Four points are concyclic if there is a cycle with .

It is so named because it passes through nine significant concyclic points defined from the triangle.

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

Most of the other results discussed in the paper pertained to various concyclic points that could be constructed from the Lemoine point.

Several other sets of points defined from a triangle are also concyclic, with different circles; see nine-point circle and Lester's theorem.

(Because of this, some authors define "concyclic" only in the context of four or more points on a circle.)

More generally, a polygon in which all vertices are concyclic is called a cyclic polygon.

This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.

Carnot's theorem is used in a proof of the Japanese theorem for concyclic polygons.

Also, it's readily shown that the quadrilateral case suffices to prove the general case of the concyclic polygon theorem.

Specifically, let be an arbitrary concyclic quadrilateral and let be the incenters of the triangles .

For n distinct points there are n(n 1)/2 bisectors, and the concyclic condition is that they all meet in a single point, the centre O.

All vertices of a regular polygon lie on a common circle (the circumscribed circle), i.e., they are concyclic points.

Triangulating an arbitrary concyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles).

Hence the concyclic polygon theorem considered here can be regarded as a corollary of the extended cyclic quadrilateral theorem.

Four Concyclic Points by Michael Schreiber, The Wolfram Demonstrations Project.

A set of five or more points is concyclic if and only if every four-point subset is concyclic.

By converse of angle in the same segment, ARBF and AFCQ are both concyclic.

Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle.

Every surjective mapping that transforms any four concyclic points into four concyclic points is a similarity.

Because BFC and BPC add up to 180 , BPCF is also concyclic.

The cross-ratio is real if and only if the four points are either collinear or concyclic, reflecting the fact that every Möbius transformation maps generalized circles to generalized circles.

A polygon which has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic).

Some authors consider collinear points (sets of points all belonging to a single line) to be a special case of concyclic points, with the line being viewed as a circle of infinite radius.

In Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter are concyclic.