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Suppose for contradiction that the critical probability for site percolation on the triangular lattice satisfied $p_c>1/2$. Then $p=1/2$ would be subcritical, so exponential decay of subcritical percolation would give constants $C,c>0$ such that the probability that a fixed site has an open path to distance $N$ is at most $C{e}^{-cN}$.

Every left-to-right open crossing of an $N$ by $N$ rhombus contains a site on its left side joined to distance at least $N$. There are $O(N)$ possible starting sites, so the union bound would implyPlanar self-duality and symmetry instead make this crossing probability exactly $1/2$ at every $N$. This contradiction proves $p_c\le 1/2$.

A closed circuit of diameter at least $n$ through a fixed listed point contains a closed path from that point to graph distance at least $n/2$. Exponential decay of subcritical percolation bounds this probability by $C{e}^{-cn}$. A union bound over the $n$ listed points givesafter reducing $u>0$ and adjusting finitely many small $n$.