A wellknown theorem in projective geometry states that the three Pascal lines of an arbitrary hexagon inscribed in a quadric intersect in one point. I found an algebraic reformulation, which states that for any six lines $l_i=a_ix+b_iy+c_iz$, if the quadrics $l_i^2$ are linearly dependent, then $l_1l_3$, $l_3l_5$, $l_1l_5$, $l_2l_4$, $l_2l_6$, $l_4l_6$ are linearly dependent. But I can not understand the algebraic meaning of this fact. Is it just an accident or a special case of some duality?
Pascal's theorem is a special case of the purely algebraic CayleyBacharach theorem. For a brief discussion see e.g. http://terrytao.wordpress.com/2011/07/15/pappusstheoremandellipticcurves/ 


You might want to look at http://ssdut.dlut.edu.cn/lzxjj/uploadfile/200905/20090512022236594.pdf (I am not certain that it answers your question...) 


Thank you very much! This is interesting article I have read it. But what I am concerned is interpretation of the result from the perspective of geometry of space of quadrics. First, I thought it follows from the polar duality in $P^5$, but it seems it does not. 

