Suppose that we have proved the Cauchy's Integral Formula for positively oriented circles, and that now we want to prove a more general version for piecewise smooth Jordan curves. This can be done by proving a statement similar to the following one:
Let $\gamma$ be a positively oriented piecewise smooth Jordan curve and $C$ a positively oriented circle in its interior cenetered at $a$. Let $f$ be holomorphic inside and on $\gamma$, possibly with a single exception at $z = a$. Then $$\int_{\gamma}f(z)\,\mathrm{d}z = \int_C f(z)\,\mathrm{d}z.$$
Now, this is usually proved by "joining" $\gamma$ and $C$ by some path(s), so that one is finally able to apply Cauchy's integral theorem (technical details seem to be unimportant here).
However, one point keeps me bothering here: I cannot prove or disprove the following conjecture:
Let $\gamma$ and $C$ be as above. Then $\gamma$ is $(G \setminus \{a\})$-homotopic to $C$, where $G$ is the union of (the image of) $\gamma$ with its interior.
Is this true? If so, then the former statement could be proved by simply applying the "Deformation theorem" asserting equality of integrals along homotopic curves.
(Seems to be related to the last comment under this question.)
One more point: I am mostly interested in an "elementary" resolution of the above question. In particular, I would prefer to avoid concepts like index or the most general form of Cauchy's theorem, which do not seem to be needed (in fact, this implies that a "better" formulation of the above statements would disregard orientation and simply assert that they are true for $\gamma$ or for $-\gamma$).