We characterise the affine span of the midpoints sets, M(x,y), for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of M(x,y) in case the associated Euclidean Jordan algebra is simple. In particular, we find for A and B in the cone positive definite Hermitian matrices that dim(affM(A,B))=q2, where q is the number of eigenvalues μ of A−1B, counting multiplicities, such that μ=max{λ+(A−1B),λ−(A−1B)−1}, where λ+(A−1B):=max{λ:λ∈σ(A−1B)} and λ−(A−1B):=min{λ:λ∈σ(A−1B)}. These results extend work by Y. Lim [18]