Cox's Theorem

Let $\sigma_1$, ..., $\sigma_4$ be four Planes in General Position through a point $P$ and let $P_{ij}$ be a point on the Line $\sigma_i\cdot\sigma_j$. Let $\sigma_{ijk}$ denote the Plane $P_{ij}P_{ik}P_{jk}$. Then the four Planes $\sigma_{234}$, $\sigma_{134}$, $\sigma_{124}$, $\sigma_{123}$ all pass through one point $P_{1234}$. Similarly, let $\sigma_1$, ..., $\sigma_5$ be five Planes in General Position through $P$. Then the five points $P_{2345}$, $P_{1345}$, $P_{1245}$, $P_{1235}$, and $P_{1234}$ all lie in one Plane. And so on.

See also Clifford's Circle Theorem