Two classes of cubic graphs are introduced. Their construction is based on the geometry of the projective line. For instance, \(T(p)\), where \(p\) is an odd prime, has as vertices the 3-subsets of \(\text{PG} (1,p)\), where \(\{a,b,c\}\) and \(\{a,b,d\}\) are adjacent when the pairs \(\{a,b\}\) and \(\{c,d\}\) harmonically separate each other. (The definition of \(H(p)\) is a bit more complex.) The graph \(T(p)\) is 2-arc-transitive. For \(p \equiv 1 \pmod 4\) it has two isomorphic connected components, while for \(p \equiv 3 \pmod 4\) it is connected. The girth of \(T(p)\) is greater than \(\log_ \varphi p\), where \(\varphi= {1+\sqrt 5 \over 2}\). The exact value for the girth of \(T(p)\) and \(H(p)\) is computed for several values of \(p\). In some cases they are the smallest known cubic graphs having that girth.