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.

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.

Angles

∠BMC

and

∠DMC

are equal.

The bisectors of the angles at $B$ and $D$ intersect on the diagonal $AC$ .

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 is located towards the sides of the quadrilateral to proportional distances to the length of these sides.

References

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

Harmonic quadrilateral

Properties

References

Further reading