> 
> If the slopes of side $AB$, $BC$, $CD$, $DA$ of a quadrilateral $ABCD$ are respectively $m\_1, m\_2, m\_3, m\_4$, and the diagonals (or the extended lines of them) are perpendicular, then the equation in $p$:
>  $$(m\_1m\_3-m\_2m\_4)p^2-((m\_2+m\_4)(1+m\_1m\_3)\\-(m\_1+m\_3)(1+m\_2m\_4))p-(m\_1m\_3-m\_2m\_4)=0$$
>  is satisfied by both slopes of the diagonals (in case both exist) or by the only existing one.
> 
> 
> 

This theorem can be proved by means of the theory of circumscribing conics (see my dummy answer below).

Does anyone know a different, more simple proof?