Clifford's Circle Theorem

Let $C_1$, $C_2$, $C_3$, and $C_4$ be four Circles of General Position through a point $P$. Let $P_{ij}$ be the second intersection of the Circles $C_i$ and $C_j$. Let $C_{ijk}$ be the Circle $P_{ij}P_{ik}P_{jk}$. Then the four Circles $C_{234}$, $C_{134}$, $C_{124}$, and $C_{123}$ all pass through the point $P_{1234}$. Similarly, let $C_5$ be a fifth Circle through $P$. Then the five points $P_{2345}$, $P_{1345}$, $P_{1245}$, $P_{1235}$ and $P_{1234}$ all lie on one Circle $C_{12345}$. And so on.

See also Circle, Cox's Theorem, Miquel's Theorem, Pivot Theorem