  A matchstick graph is a plane graph with edges drawn as unit-distance line
segments. Harborth introduced these graphs in 1981 and conjectured that the
maximum number of edges for a matchstick graph on $n$ vertices is $\lfloor
3n-\sqrt{12n-3} \rfloor$. In this paper we prove this conjecture for all $n\geq
1$. The main geometric ingredient of the proof is an isoperimetric inequality
related to L'Huilier's inequality.
