# Projective plane and some curves

We define a line in the projective plane as a set of the form $$L_{a,b,c} = \left\{ {\left[ {x,y,z} \right] \in P_R^2 :ax + by + cz = 0} \right\}\text{ or just }L$$ Let a finite collection of lines $$\left\{ {L_i } \right\}_{i = 1}^n$$ such that $$\bigcap\limits_{i = 1}^n {L_i }$$ it´s empty The last definition.. a point $$p \in \left\{ {L_i } \right\}_{i = 1}^n$$ it´s said to be a k-point if it´s exactly contained in k lines of $$\left\{ {L_i } \right\}_{i = 1}^n$$ Prove that: $$t_2 \geqslant 3 + \sum\limits_{k \geqslant 4} {\left( {k - 3} \right)t_k }$$ where $t_k$ is the amount of k-points. And the equality holds iff the corresponding paving is by triangles ( every such kind of sets define a natural paving by polygons)

This problem looks so difficult, I have no idea how to attack this problem Dx This problem scares me , if someone can help me )=

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If you remove the «homework difficult problem?» from the title nothing is lost... – Mariano Suárez-Alvarez♦ Dec 4 '11 at 7:00

## 1 Answer

Look into what is known as the Sylvester-Gallai Theorem, and combinatorial approaches to proving it.

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