  We pose and study the Fermat-Torricelli problem for a triangle in the
projective plane under the sine distance. Our main finding is that if every
side of the triangle has length greater than $\sin 60^\circ$, then the
Fermat-Torricelli point is the vertex opposite the longest side. Our proof
relies on a complete characterization of the equilateral case together with a
deformation argument.
