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If a function $f$ is analytic in an open set $U$, then $\int_{\partial T}f(z)dz=0$ for every closed triangle $T$ in $U$.

I already have the result for a rectangle but how can I prove that for a triangle you also have this? What other geometric figures do you have this result for? Could anyone help me, please? Thank you very much.

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    $\begingroup$ Under the assumption that $U$ is simply connected, Cauchy's theorem gives you that the integral along any "nice" (rectifiable) loop in $U$ is 0. $\endgroup$
    – Bass
    Nov 1, 2017 at 19:30
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    $\begingroup$ See math.stackexchange.com/a/505939/589. $\endgroup$
    – lhf
    Nov 1, 2017 at 19:43

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It holds for any polygon. Can you see why from this picture?

enter image description here

You can approximate any closed curve arbitrarily well with a polygon, so it actually holds for all closed curves "which are boundaries of subsets of $U$". (This is made precise by considering closed loops that are null-homotopic)

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