Another way to state your condition is to say that every 2 dimensional subspace is norm one complemented. This implies that the space is isometrically isomorphic to a Hilbert space. I am not sure who first proved this, but it is a remark at the end of Bruck's paper

http://www.jstor.org/stable/pdfplus/2039349.pdf?acceptTC=true

Another way to state your condition is to say that every 2 dimensional subspace is norm one complemented. This implies that the space is isometrically isomorphic to a Hilbert space. Kakutani proved this in Some characterizations of Euclidean space, Jap. J. Math. 16, (1939). 93–97. It is also a remark at the end of Bruck's paper

http://www.jstor.org/stable/pdfplus/2039349.pdf?acceptTC=true