Exercise 3.4 in Rotman's An Introduction to Algebraic Topology

I am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution.

Exercise 3.4:

Let $\sigma:\Delta^2 \to X$ be continuous, where $\Delta^2=[e_0,e_1,e_2]$. Define $\varepsilon_0:I \to \Delta^2$ as the affine map with $\varepsilon_0(0)=e_1$ and $\varepsilon_0(1)=e_2$; similarly, define $\varepsilon_1$ by $\varepsilon_1(0)=e_0$ and $\varepsilon_1(1)=e_2$, and define $\varepsilon_2$ by $\varepsilon_2(0)=e_0$ and $\varepsilon_2(1)=e_1$. Finally, define $\sigma_i=\sigma \circ \varepsilon_i$ for $i=0,1,2$.

1. Prove that $(\sigma_0 \ast \sigma_1^{-1}) \ast \sigma_2$ is null homotopic rel $\{0,1\}$.
2. Prove that $(\sigma_1 \ast \sigma_0^{-1}) \ast \sigma_2^{-1}$ is null homotopic rel $\{0,1\}$.
3. Let $F:I \times I \to X$ be continuous, and define paths $\alpha, \beta, \gamma, \delta$ in $X$ by $\alpha(t)=F(t,0), \beta(t)=F(t,1), \gamma(t)=F(0,t) $, and $\delta(t)=F(1,t)$. Prove that $\alpha \simeq \gamma \ast \beta \ast \delta^{-1}$ rel $\{0,1\}$.

As a hint, Rotman suggests the use of Theorem 1.6, which states that for a continuous map $f:S^n \to Y$, the following are equvalent:

1. $f$ is nullhomotopic;
2. $f$ can be extended to a continuous map $D^{n+1} \to Y$;
3. if $x_0 \in S^n$ and $k:S^n \to Y$ is the constant map at $f(x_0)$, then there is a homotopy $F:f \simeq k$ with $F(x_0,t)=f(x_0)$ for all $t \in I$.

I'm not sure how the hint applies to the exercise. Neither do I see how to proceed without the hint. I can write down explicit formulas for $\varepsilon_i, i=0,1,2$, but I'm not sure where to go from there.

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