Ellipsoid characterization theorems

In this note we prove two ellipsoid characterization theorems. The first one is that if $K$ is a convex body in a normed space with unit ball $M$, and for any point $p \notin K$ and in any 2-dimensional plane $P$ intersecting $\inter K$ and containing $p$, there are two tangent segments of the same normed length from $p$ to $K$, then $K$ and $M$ are homothetic ellipsoids. Furthermore, we show that if $M$ is the unit ball of a strictly convex, smooth norm, and in this norm billiard angular bisectors coincide with Busemann angular bisectors or Glogovskij angular bisectors, then $M$ is an ellipse.