EGMO 2014 Problem 1

Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.

Proposed by S. Khan, UNK The answer is the interval $[2 / 3,2]$.