Put H is intersection of AB and CD

\(\Rightarrow\left\{{}\begin{matrix}\left\{{}\begin{matrix}AC^2=HC^2+AH^2\\AD^2=HD^2+AH^2\end{matrix}\right.\Rightarrow AC^2-AD^2=\left(HC^2+AH^2\right)-\left(HD^2+AH^2\right)=HC^2-HD^2\\\left\{{}\begin{matrix}BC^2=HC^2+BH^2\\BD^2=HD^2+BH^2\end{matrix}\right.\Rightarrow BC^2-BD^2=\left(HC^2+BH^2\right)-\left(HD^2+BH^2\right)=HC^2-HD^2\end{matrix}\right.\)

=> AC² - AD² = BC² - BD²