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.

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.
• The point of intersection of the diagonals is located towards the sides of the quadrilateral to proportional distances to the length of these sides.

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  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.
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  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.