Harmonic quadrilateral explained

In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle, 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 be a harmonic quadrilateral and the midpoint of diagonal . Then:

• Tangents to the circumscribed circle at points and and the straight line either intersect at one point or are parallel. Therefore, the pole of each diagonal is contained in the other diagonal respectively.^[1]
• Angles and are equal.
• The bisectors of the angles at and intersect on the diagonal .
• A diagonal of the quadrilateral is a symmedian of the angles at and in the triangles ∆ and ∆.
• 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.^[2]
• Considering the points,,, as complex numbers, the cross-ratio .^[3]

Further reading

• Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913.

Notes and References

1. Web site: Some Properties of the Harmonic Quadrilateral. Proposition 7
2. Web site: Some Properties of the Harmonic Quadrilateral. Proposition 6
3. Web site: HarmonicQuad.