[math-fun] Neat Dragon question

What is the longest line segment inscribable in the Dragon linking 0+0i to 1 + 0i via (1+i)/2? "Hint": If the segment be t z2 + (1-t)z1, with 0<t<1, then every point on it is (empirically, at least) a triple point of the spacefill whenever t is a dyadic rational! —rwg (This impacts OEIS A260747, etc.)