Harmonic quadrilateral

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.
  • A diagonal BD of the quadrilateral is a symmedian of the angles at B and D in the triangles ∆ABC and ∆ADC.


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

Further reading

  • Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913.
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