The equal tangents property
  Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane then M is an Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different. 
J. Jeronimo-Castro, G. Ruiz-Hernandez, S. Tabachnikov
math.MG