The quantum steering ellipsoid formalism naturally extends the Bloch vector picture to provide a visualisation of two-qubit systems. If Alice and Bob share an entangled state then a local measurement by Bob steers Alice’s Bloch vector; given all possible measurements by Bob, the set of states to which Alice can be steered forms her steering ellipsoid inside the Bloch sphere. This gives us a novel geometric perspective on a number of quantum correlation measures such as entanglement, CHSH nonlocality and singlet fraction. In particular, by analysing a tripartite scenario we find that steering ellipsoid volumes obey a simple monogamy relation from which one can derive the well-known CKW (Coffman-Kundu-Wootters) inequality for the monogamy of entanglement. Remarkably, we can also use steering ellipsoids to derive some highly non-trivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.