In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle,^[1] is a quadrilateral that can be inscribed in a circle (cyclic quadrilateral) 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 parallel. Therefore, the pole of each diagonal is contained in the other diagonal respectively.^[2][3]
• 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.
• The point of intersection of the diagonals is located towards the sides of the quadrilateral to proportional distances to the length of these sides.
• The point of intersection of the diagonals minimizes the sum of squares of distances from a point inside the quadrilateral to the quadrilateral sides.^[4]
• Considering the points A, B, C, D as complex numbers, the cross-ratio (ABCD) = −1.^[3]

• 
  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.

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