Cyclic quadrangle in which the products of opposite side lengths are equal
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
diagonalAC. 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.
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,
ISBN978-0-486-46237-0
Further reading
Gallatly, W. "The Harmonic Quadrilateral." §124 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 90 and 92, 1913.