However, instead of requiring that the products of opposite sides are equal, we can stipulate that the products of consecutive sides are equal, and then call the resulting cyclic quadrilateral quasi harmonic.
Having this latter requirement (e.g. in the diagram below) yields some nice lateral properties for quadrilaterals that satisfy them.
Any rectangle can be inscribed in a circle, and the products of consecutive sides are equal (e.g. below). Thus, rectangles are quasi-harmonic as per our definition. Note that a rectangle is not a harmonic quadrilateral, unless the rectangle is square.
Right kites are both harmonic and quasi-harmonic. So are squares.





We need to show that it’s cyclic, and that the products of the lengths of consecutive sides are equal.
- side-lengths:
,
,
,
- diagonals:
,
- Ptolemy’s theorem:
, and
. Confirms
is cyclic.
- product of consecutive sides:
. Quasi harmonic.





- side-lengths:
,
,
,
- diagonals:
,
- Ptolemy’s theorem:
, and
. Confirms
is cyclic.
- product of consecutive sides:
. Quasi harmonic.





- side-lengths:
,
,
,
- diagonals:
,
- Ptolemy’s theorem:
, and
. Confirms
is cyclic.
- product of consecutive sides:
. Quasi harmonic.
Takeaway
Let be a convex cyclic quadrilateral with side-lengths
,
,
, and
. Then the following statements are equivalent:
- diagonal
bisects diagonal
.
The longer diagonal in a quasi harmonic quadrilateral always bisects the shorter diagonal.
Task
- (Bisection condition) Let
be a convex cyclic quadrilateral with side-lengths
,
,
, and
. PROVE that:
- diagonal
bisects diagonal
if and only if
- diagonal
bisects diagonal
if and only if
- diagonals
and
bisect each other if and only if
and
(rectangle).
- diagonal
- (Basic characteristics) Let
be a convex cyclic quadrilateral in which the side-lengths are
,
,
,
, and the diagonals are
,
. If
(quasi-harmonic), PROVE that:
-
.