Suppose we have $n$ points in a plane, not all on the same line. Then they are determining at least $n$ different lines.
Suppose points $T_1,...,T_n$ determine $m$ lines $$\ell_i(x,y) :\;\;a_ix+b_iy+c_i=0;\;\;\;1\leq i\leq m$$ and let $L_k$ be a set of all lines $\textbf{not}$ through $T_k$ and let $$p_k(x,y) = \prod _{\ell_i\in L_k}\ell_i(x,y)$$ Then, since no line contains all points, the degree of $p_k\leq m-1$. Also we have $p_k(T_k) \neq 0$ and $p_j(T_k) =0$ for all $j\ne k$ which means that all $n$ polynomials are independent over vector space $\mathbb{R}_{m-1}[x,y]$ which has a dimension ${m+1\choose 2}$. So $$n\leq {m+1\choose 2} \implies m\geq {\sqrt{8n+1}+1\over 2}$$ which is pretty weak bound.
Any idea how to improve this approach?
Edit: source