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.

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.

References

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

Further reading