When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is right. I am looking for a proof of the conjecture.

Conjecture:Let four points $P_1, P_2, P_3, P_4$ on a plane $\sigma$. Let six planes $\sigma_{12}$, $\sigma_{13}$, $\sigma_{14}$, $\sigma_{23}$, $\sigma_{24}$ such that the plane $\sigma_{ij}$ through two points $P_i$ and $P_j$. Let $P_{ijk}$ be a common points of three planes $\sigma_{ij}, \sigma_{jk}, \sigma_{ki}$. Then show that four points $P_{123}, P_{124}, P_{134}, P_{234}$ lie on a plane.

Alternatively, take a purely incidence proof of Cox's theorem and dualize the proof. Again, this is well-known projective geometry, not research: please ask on math.stackexchange.com for more explanations on projective duality if you need, but not here. $\endgroup$ – Gro-Tsen Mar 30 '16 at 13:39