  A metric polygon is a metric space comprised of a finite number of closed
intervals joined cyclically. The second-named author and Ntalampekos recently
found a method to bi-Lipschitz embed an arbitrary metric triangle in the
Euclidean plane with uniformly bounded distortion, which we call here the
tripodal embedding. In this paper, we prove the sharp distortion bound
$4\sqrt{7/3}$ for the tripodal embedding. We also give a detailed analysis of
four representative examples of metric triangles: the intrinsic circle, the
three-petal rose, tripods and the twisted heart. In particular, our examples
show the sharpness of the tripodal embedding distortion bound and give a lower
bound for the optimal distortion bound in general. Finally, we show the
triangle embedding theorem does not generalize to metric quadrilaterals by
giving a family of examples of metric quadrilaterals that are not bi-Lipschitz
embeddable in the plane with uniform distortion.
