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

Harmonic quadrilateral

Properties

References

Further reading