Show that the following two statements are equaivalent: (i) any closed path in $D$ is homotopic to zero, and (ii) any two paths in D that have common ends are homotopic to each other.
My approach is: Let $D \subset \mathbb C$ be a domain and $\gamma : [\alpha, \beta] \to D$ a closed path. Let $a \in D$ be a given point. Assume that $\gamma$ is homotopic to that point $a$. Is it enough to prove that $\gamma$ is homotopic to a point $b$ in $D$ for any $b \in D$.
Want to show that $\gamma_a(t) = a$ is homotopic to $\gamma_b(t)=b$. Then the result would follow, since homotopy is a equivalence relation:
Let $H(t,s) = (1-s)a+sb$. $H$ is continuous, and $H(t,0) = a$, $H(t,1) = b$, and $H(\alpha,s) = H(\beta,s)$. The problem is how to continue from there.