Rotate the position so that $L$ is the $x$-axis, for ease of exposition. We can forget points $d$, $e$, and $f$, and look for $x$ which minimises the sum of the distances from $(x,0)$ to $a$, $b$, and $c$. If $x$ varies by $\delta x$, then the distance from $(x,0)$ to $a$ (say) varies by $-\delta x \cos A$, where $A$ is the angle that the line from $a$ to $(x,0)$ makes with the positive $x$-axis. So we want the point $(x,0)$ such that $\cos A + \cos B + \cos C = 0$.
I can't see an explicit formula for $x$ (though I won't be surprised if somebody else can). But you can find this point to any required accuracy by a simple binary search. If you need a faster method, then Newton will help.