# A conjecture on six planes [closed]

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.

• The dual of the statement contained in the first half of the MathWorld article "Cox's Theorem" is: «Let $P_1,\ldots,P_4$ be four points in general position on a plane $σ$ and let $σ_{ij}$ be a plane on the line $P_i P_j$. Let $P_{ijk}$ denote the intersection of $σ_{ij},σ_{ik},σ_{jk}$. Then the four points $P_{234},P_{134},P_{124},P_{123}$ are all on one plane $σ_{1234}$.» In what way is this different from the "conjecture" you wrote? Even the notation is the same! – Gro-Tsen Mar 30 '16 at 12:11