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 mutually parallel.
• 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.

Further reading

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