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).

- (Basic characteristics) Let be a convex cyclic quadrilateral in which the side-lengths are , , , , and the diagonals are , . If (quasi-harmonic), PROVE that:
- .