Symmedian Point

The centroid $G$ and the symmedian point $K$ have barycentric coordinates $1:1:1$ and $a^2:b^2:c^2$ respectively, and so the line $GK$ has barycentric equation $(b^2-c^2)x + (c^2-a^2)y + (a^2 - b^2)z \; = \; 0$ while the line $AB$ has barycentric equation $z \; = \; 0$ These two lines meet at the point with baycentric coordinates $a^2-c^2:b^2-c^2:0$ . Since $AB$ and $GK$ are parallel, this must be the point at infinity, and hence its coordinates must sum to $0$ . Thus we deduce that $a^2 + b^2 = 2c^2$ . With $a=7$ and $c=13$ , we deduce that $b = \boxed{17}$ .

Let $CK^\to \cap AB=N$ and $CG^\to \cap AB = M$ . Since $GK || MN$ the Basic Proportionality Theorem (or Thales' Theorem) gives: $\frac{CG}{GM}=\frac{CK}{KN}$ The centroid splits the median in ratio $2:1$ from the vertex and the symmedian point splits the symmedian from vertex $C$ in ratio $\frac{a^2+b^2}{c^2}$ . Substituting these values in the result form Thales' Theorem and rearranging gives: $2c^2=a^2+b^2$ $\Rightarrow b=\sqrt{2c^2-a^2}=\sqrt{2.13^2-7^2}=\fbox{17}$