An augmented simplicial cone has an explicit chain contraction

Adjoining $a$ stays inside $K$ by the cone hypothesis. Permuting the original vertices changes $[a,v_0,\dots ,v_n]$ by the same sign, so $h$ respects the oriented-chain relations. If $a$ is absent from $s=[v_0,\dots ,v_n]$, expansion gives $\mathrm{\partial }[a,v_0,\dots ,v_n]=s+{\sum }_{i=0}^n(-1{)}^{i+1}[a,v_0,\dots ,{\widehat{v}}_i,\dots ,v_n]=s-h\mathrm{\partial }s$.

If $a=v_j$, then $h(s)=0$. In $h\mathrm{\partial }s$ all terms except deletion of $v_j$ repeat $a$ and vanish. The remaining term is $(-1{)}^j[a,v_0,\dots ,{\widehat{v}}_j,\dots ,v_n]=s$, because moving $a$ back to position $j$ contributes another $(-1{)}^j$. In degree zero this says $\mathrm{\partial }[a,v]+[a]=[v]$ for $v\mathrm{\ne }a$, and $0+[a]=[a]$ for $v=a$. In degree $-1$, $\mathrm{\partial }h(1)=\mathrm{\partial }[a]=1$. Thus the identity holds on all generators and hence all chains.

The identity $1=\mathrm{\partial }h+h\mathrm{\partial }$ is a null homotopy of the identity, hence contractibility. A chain of faces of a nonempty full simplex can always be enlarged by the maximal face; thus its order complex, as defined in Barycentric subdivision of an abstract simplicial complex, is a cone with that face as specified apex, and the same calculation applies.