I'm reading Complex Function Theory by Palka.
Given a closed, piecewise smooth curve $\gamma:[a,b] \to \mathbb{C}$, its winding number about $z_0 \in \mathbb{C}$ (which doesn't intersect the curve) is $$n( \gamma,z_0):= \frac{1}{2 \pi i}\int_\gamma \frac{dz}{z-z_0}$$ I would like to show from first principles that if $T \subseteq \mathbb{C}$ is a triangle with vertices a,b,c (in counterclockwise (CCW) order), and $\partial T$ is the closed, piecewise smooth curve on the boundary of T going CCW (from a to b to c and back to a again), then $n( \partial T,0)=1$. (Here, WLOG, $0 \in T^o$, the interior of T)

I know of (but am not sophisticated enough to understand the proof of) the Jordan Curve Thm, so I don't want to simply cite that.
Palka also mentions the Cauchy integral formula, from which my question also follows, but again to use that result would be to assume $n( \partial T,0)=1$.
In the below image, Palka shows the result for rectangles. I tried to copy the proof for triangles, but my issue is if you inscribe a triangle in a circle, the center of the circle might not be in the triangle, so I ran into difficulty.

Thanks a lot in advance! enter image description here


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