A=(0,0); B=(√3/2,-1/2), C=(√3/2,1/2), D=(√3,0); second copy rotated about A by θ, cosθ=5/6, sinθ=√11/6, so sin²(θ/2)=1/12 and |DD'|=2√3·sin(θ/2)=1 exactly. Code below is a floating-point filter (tol 1e-12) plus exhaustive colouring search; the exact argument is the hand derivation.