Let $T$ be a (closed) triangle in $\mathbb{R}^{2}$ with edges $I, J, K$. Let $A, B, C$, be closed subsets of $T$, such that $I \subset A, J \subset B, K \subset C$ and $T=A \cup B \cup C$. Prove that $A \cap B \cap C$ is non-empty.

Deduce that there is no continuous map $f: D \rightarrow \partial D$ such that $f(p)=p$ for all $p \in \partial D$, where $D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leqslant 1\right\}$ is the closed unit disc and $\partial D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is its boundary.

Let now $\alpha, \beta, \gamma \subset \partial D$ be three closed arcs, each arc making an angle of $2 \pi / 3$ (in radians) in $\partial D$ and $\alpha \cup \beta \cup \gamma=\partial D$. Let $P, Q$ and $R$ be open subsets of $D$, such that $\alpha \subset P$, $\beta \subset Q$ and $\gamma \subset R$. Suppose that $P \cup Q \cup R=D$. Show that $P \cap Q \cap R$ is non-empty. [You may assume that for each closed bounded subset $K \subset \mathbb{R}^{2}, d(x, K)=\min \{\|x-y\|: y \in K\}$ defines a continuous function on $\mathbb{R}^{2}$.]