Abstract: Some very non-trivial results in classical Euclidean elementary geometry can be derived using the steering ellipsoid formalism from quantum mechanics. I will outline the theory behind steering ellipsoids and discuss their significance within the broad field of quantum information theory. A geometric understanding of quantum entanglement gives a novel derivation of the famous Euler inequality in 2 and 3 dimensions: given one circle (sphere) inside another circle (sphere), when does there exist a triangle (tetrahedron) circumscribed about the smaller and inscribed in the larger? Remarkably, the steering ellipsoid approach also gives a new inequality that applies to the more general case of ellipses and ellipsoids.