= Solution
<Cauchy's integral theorem> states that if a function is holomorphic on a simply connected open set, its integral around every closed piecewise-smooth path in that set is zero. A local form sufficient here is: if a function is holomorphic on an open set containing a triangle together with its interior, then
$$
\boxed{\int_{\partial\Delta}f(z)\,dz=0.}
$$
For a general domain with holes, an arbitrary closed path need not have zero integral; the triangle or simple-connectedness hypothesis supplies the required interior.
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