Lef $f,g: I \rightarrow I^2$ be continuous and $f(0) = (a,0), f(1) = (b,1)$ and $g(0) = (0,c), g(1) = (1,d)$. Prove that there exist $t_1, t_2 \in I$ such that $f(t_1)=g(t_2)$
The book is hinting to use (if $h : D^n \rightarrow D^n$ continuously then there exists a fixed point) to find a function from $I^2 \rightarrow I^2$, but I can't find anything that works even if I replace $D^n$ with $I^n$ by homeomorphism.