ISL 1988 Problem 89

We match sets $M$ of points in the coordinate plane to sets $M*$ according to the rule that $(x*,y*) \in M*$ if and only if $x \cdot x* + y \cdot y* \leq 1$ whenever $(x,y) \in M.$ Find all triangles $Q$ such that $Q*$ is the reflection of $Q$ in the origin.