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* Angles and are equal.
* The bisectors of the angles at and intersect on the diagonal .
* A diagonal of the quadrilateral is a 
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, a harmonic quadrilateral, or harmonic quadrangle, is a quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
that can be inscribed in a circle ( cyclic quadrangle) in which the products of the lengths of opposite sides are equal. It has several important properties.
Properties
Let be a harmonic quadrilateral and themidpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
of diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
. Then:
* Tangents to the circumscribed circle at points and and the straight line either intersect at one point or are mutually parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
.
* Angles and are equal.
* The bisectors of the angles at and intersect on the diagonal .
* A diagonal of the quadrilateral is a symmedian
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the ...
of the angles at and in the triangles ∆ and ∆.
* The point of intersection of the diagonals is located towards the sides of the quadrilateral to proportional distances to the length of these sides.
References
Further reading
* Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913. Types of quadrilaterals {{geometry-stub