# 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.

## 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.

## References

- 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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