Let $A,B,C,D$ be points. Prove that $AB\cap CD=\frac{(\overline{a}b-a\overline{b})-(a-b)(\overline{c}d-c\overline{d})}{(\overline{a}-\overline{b})(c-d)-(a-b)(\overline{c}-\overline{d})}$. (Here the lower case letter denotes the complex number representing the upper case point).
Does anyone have a proof for this?
If we let $AB\cap CD=k$, I can in fact verify that this is the answer since I can check that $\det{\begin{vmatrix} a & \overline{a} & 1 \\ k & \overline{k} & 1 \\ b & \overline{b} & 1 \\ \notag \end{vmatrix}}=0$ and that $\det{\begin{vmatrix} c & \overline{c} & 1 \\ k & \overline{k} & 1 \\ d & \overline{d} & 1 \\ \notag \end{vmatrix}}=0$ (although it is a lot of computation), but what is the way to derive the formula?