= Solution
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 $Ce^{-cN}$.
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 imply
$$
\mathbb P_{1/2}(\text{left-to-right crossing})
\leq C'Ne^{-cN}\longrightarrow0.
$$
Planar self-duality and symmetry instead make this crossing probability exactly $1/2$ at every $N$. This contradiction proves $p_c\leq1/2$.
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