Show $f$ has a fixed point if $f\simeq c$ I have the following problem: 
Show that if $f:S^1\to S^1$ is a continuous map, and $f$ is homotopic to a constant, then $\exists p\in S^1 : f(p)=p$.
My approach is to show that if for all $p, \ $ $f(p)\neq p$, then $f$ is homotopic to $\mathrm {id}_{S^1}$. To prove this I thought of using the parametrization of the segment $pf(p)$, and projecting outwards to $S^1$. However this creates problems if the points are antipodes. I suppose then, that this assumption is too strong, and I should either prove that $f$ is homotopic to $z^n$ (by contradiction) for non zero $n$, or bring in some theorem such as Borsak-Ulam, which applies here because the map is non-surjective, and can be thought of as a map into $\mathbb R$, but I can't think of how this would help. I would appreciate some help.

An alternative argument would be to show that if $f$ has no fixed point, then $f$ is homotopic to the antipodal map $h(x)=-x$. 
