Harmonic quadrilateral
In Euclidean geometry, a harmonic quadrilateral is a quadrilateral whose four vertices lie on a circle, and whose pairs of opposite edges have equal products of lengths.
Harmonic quadrilaterals have also been called harmonic quadrangles. They are the images of squares under Möbius transformations. Every triangle can be extended to a harmonic quadrilateral by adding another vertex, in three ways. The notion of Brocard points of triangles can be generalized to these quadrilaterals.
Definitions and characterizations
A harmonic quadrilateral is a quadrilateral that can be inscribed in a circle and in which the products of the lengths of opposite sides are equal. Equivalently, it is a quadrilateral that can be obtained as a Möbius transformation of the vertices of a square, as these transformations preserve both the inscribability of a square and the cross ratio of its vertices. Four points in the complex plane define a harmonic quadrilateral when their complex cross ratio is ; this is only possible for points inscribed in a circle, and in this case, it equals the real cross ratio.Constructions
For any point in the plane, the four lines connecting to each vertex of the square cut the circumcircle of the square in the four points of a harmonic quadrilateral.Every triangle can be extended to a harmonic quadrilateral in three different ways, by adding a fourth vertex to the triangle, at the point where one of the three symmedians of the triangle cross its circumcircle. Each symmedian is the line through one vertex of the triangle and through the crossing point of the two tangent lines to the circumcircle at the other two vertices.