= Solution
Inside the <triangle>, both $x$ and $y$ are nonzero. Thus $\nabla f=0$ reduces to $1-2x+y=0$ and $1-x+2y=0$, whose unique solution is $(1/3,-1/3)$. At that point
$$
H=\begin{pmatrix}2/3&-1/3\\-1/3&2/3\end{pmatrix},\qquad f=-\frac1{27}.
$$
The <Hessian matrix> has positive <eigenvalues> $1$ and $1/3$, so this is a strict local <minimum>. It is also the minimum on the closed <triangle>: $f=0$ on the boundary, $f<0$ inside, and there is no other interior <stationary point>. Therefore
$$
\boxed{(x,y)=(1/3,-1/3),\qquad f_{\min}=-1/27.}
$$
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