Harmonic quadrilateral

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 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.

References

^ Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover, p. 100, ISBN 978-0-486-46237-0
^ "Some Properties of the Harmonic Quadrilateral". Proposition 7
^ ^a ^b "HarmonicQuad".
^ "Some Properties of the Harmonic Quadrilateral". Proposition 6

Properties

References

Further reading