Is the Torricelli/Fermat point unique? In a triangle with all angles less than 120 degrees can there be two such points? I am wondering because in the geometry game _Euclidea_ it gives two "v-stars" for this problem (theta 8.6). This usually means that there are two solutions possible. I found one solution, basically by constructing equilateral triangles on the sides and connecting their external vertices. However I am confused what the second solution is supposed to be.

There is one Fermat–Torricelli point, i.e. the point such that the total distance from the three vertices of the triangle to the point is the minimum possible

There are two isogonic centres of a triangle (except for special cases where one of the original angles is $120^\circ$ or $60^\circ$ such as an equilateral triangle) which form angles of $120^\circ$ or $60^\circ$ with each of the three pairs of vertices of the original triangle

When all the original angles are less than $120^\circ$ then the Fermat–Torricelli point is one of the isogonic centres (the one inside the original triangle). When one of original angles is greater $120^\circ$ then that vertex is the Fermat–Torricelli point while both isogonic centres are outside the original triangle

So perhaps if something suggests there are two solutions to this type of question, it is referring to isogonic centres rather than Fermat–Torricelli points