On Bretschneider's generalization of Ptolemy's theorem Mathematician Carl Anton Bretschneider (1808–1878), from Gotha (Germany), carried out the following generalization of Ptolemy’s theorem, which refers to the product of diagonals in a convex quadrilateral.

Assume that ABCD is a convex quadrilateral. Then
$$AC^2\cdot{
BD^2} = AB^2\cdot{CD^2} + AD^2\cdot{BC^2} − 2\cdot{AB}\cdot{BC}\cdot{CD}\cdot{DA}\cdot{cos{(\angle{B} + \angle{D})}}$$

is valid, where $\angle{B}$ and $\angle{D}$ are the inner angles at the vertices of the given quadrilateral.
I have taken this generalization from this article, but it does not cite the source. So my question is: what is the original source of this generalization?

The source appears to be Bretschneider, C. A. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Math. 2, 225-261, 1842.
The equation you're asking about is the first equation of group 32 on pg 237.

The terms are defined on pg 227, except for $\psi$, which is defined on pg 234.
The paper presents a large number of formulae in a similar vein that relate to the so-called complete quadrangle.
