Editing Concyclic points (section)

=== Other concyclic points ===

In any triangle all of the following nine points are concyclic on what is called the [[nine-point circle]]: the midpoints of the three edges, the feet of the three [[altitude (geometry)|altitudes]], and the points halfway between the [[orthocenter]] and each of the three vertices.

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

If [[Line (mathematics)|lines]] are drawn through the [[Lemoine point]] [[parallel (geometry)|parallel]] to the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the [[Lemoine circle]].

The [[van Lamoen circle]] associated with any given triangle <math>T</math> contains the [[circumcenter]]s of the six triangles that are defined inside <math>T</math> by its three [[median (geometry)|median]]s.

A triangle's [[circumcenter]], its [[Lemoine point]], and its first two [[Brocard points]] are concyclic, with the segment from the circumcenter to the Lemoine point being a [[diameter]].<ref>Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry", ''[[Mathematical Gazette]]'' 83, November 1999, 472–477.</ref>