If every two-dimensional (vector) subspace of a normed space is an inner product space, then so is that normed space
Since $|a+b|^2+|a-b|^2=2(|a|^2+|b|^2)$ is trivial when $a\parallel b$, and otherwise $a,\,b$ span a $2$-dimensional subspace of $X$, the identity is true in general, and the inner product on $X$ is then defined by a suitable polarization identity.