My definition is that a space $X$ is contractible if it is homotopy equivalent to a point, i.e. there exists $f:X\rightarrow\{pt\}$ and $g:\{pt\}\rightarrow{X}$ such that $f\circ{g}\simeq{id}_{\{pt\}}$ and $g\circ{f}\simeq{id}_X$. I see all over the place (without proof) that a space is contractible if and only if its identity map is nullhomotopic, i.e. there exists a homotopy $F:X\times{I}\rightarrow{X}$ such that $F(x,0)=id_X$ and $F(x,1)$=constant.
I have seen many other statements which would imply this - such as a space $X$ is contractible if and only if every map $f:X\rightarrow{Y}$, for arbitrary $Y$, is nullhomotopic - but all seem to use the statement above as part of the proof. I feel like it should be easy but have got nowhere.