projective plane and tetrahedra

suppose we have a tetrahedron with four adjacent ones in RP3 and the vertices of the central tetrahedron are [1,0,0,0,0] , [0,1,,0,0] , [0,0,1,0] , [0,0,0,1]. For each vertex does the collection of lines passing through the vertex form a projective plane? why?

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The collection of lines through any point of RP3 form a RP2 (just look in coordinates and you see it). I dont see the relation with the tetrahedron. What did you have in mind when speaking about the tetrahedron and the set of lines? – Jérémy Blanc Feb 2 at 12:27
suppose we have the point [1,0,0,0] in RP3 and the lines going through this point . I just can't see why this collection form a projective plane. – DAVID Feb 3 at 3:17
Take coordinates [w,x,y,z] on RP3. Any line through [1,0,0,0] intersects the plane w=0 into exactly one point, and each point $p$ of this plane gives an unique line, which is the line through [1,0,0,0] and $p$. This plane is a projective plane, so you get the parametrisation. – Jérémy Blanc Feb 3 at 12:01
is it correct to say that we are using the ' principle of duality ' here ? – DAVID Feb 4 at 4:29
Voted to close; perhaps math.stackexchange.com is a better place for such discussion. – quid Feb 4 at 22:25