In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle,[1] is a quadrilateral 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.


Let ABCD be a harmonic quadrilateral and M the midpoint of diagonal AC. Then:

Tangents to the circumscribed circle at points A and C and the straight line BD either intersect at one point or are mutually parallel.
Angles ∠BMC and ∠DMC are equal.
The bisectors of the angles at B and D intersect on the diagonal AC.
The point of intersection of the diagonals is located towards the sides of the quadrilateral to proportional distances to the length of these sides.


  1. ^ Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover, p. 100, ISBN 978-0-486-46237-0

Further reading