Harmonic quadrilateral

A harmonic quadrilateral is a quadrilateral that can be inscribed in a circle (a cyclic quadrilateral) and in which the products of the lengths of opposite sides are equal (an Apollonius quadrilateral). 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 -1; this is only possible for points inscribed in a circle, and in this case, it equals the real cross ratio. ==Constructions==

For any point p in the plane, the four lines connecting p to each vertex of the square cut the circumcircle of the square in the four points of a harmonic quadrilateral. ==Properties==

The definition of the Brocard points of a triangle can be extended to harmonic quadrilaterals. A Brocard point of a polygon has the property that the line segments connecting the Brocard to the polygon vertices all form equal angles with the adjacent polygon sides. Each triangle has two Brocard points, one that forms equal angles with the polygon sides adjacent in the clockwise direction from each vertex, and another for the counterclockwise direction. The same property is true for the harmonic quadrilaterals, uniquely among cyclic quadrilaterals. ==See also==