Arrangements of n Points whose Incident-Line-Numbers are at most n/2

We consider a set X of n points in the plane, not all in a line, and the set of lines L spanned by X, where we say that a line is spanned by X if it passes through at least two elements of X. For a point P of X, we denote by t(P ) the number of lines in L which are incident to P , and call it the incident-line-number of P . We define t(X) to be MaxP∈Xt(P ), and call it the maximum incident-line-number of X. What values does t(X) take for various arrangements X of n points? If X is an n point set, any three of them are not on a line, then we have t(X) = n− 1. So the maximum value for t(X) is n − 1. What is the minimum value for t(X)? Dirac showed the following inequality for t(X) and posed a conjecture.