geometry
The Fermat–Torricelli point of a triangle is a point which minimizes the total distance from the point to the vertices.
The geometric method of finding the Fermat–Torricelli point for triangles is well known.
We may apply Lagrange Multipliers to find such a point for polygons.
Is there a geometric construction of Fermat–Torricelli point for polygons ??
Let $C_i$ convex sets. For any points $a_i,b_i\in C_i$ the segment $a_ib_i$ is contained on $C_i$. So if $p,q\in C_1\cap C_2$ then the segment $pq$ is contained on $C_1\cap C_2$ also. So the intersection is again a convex set.
The above picture depicts the Fermat-Torricelli point. From there we see that $$\begin{aligned}S^2 &= b^2+c^2-2bc\cos(A+\frac\pi3)\\ &=b^2+c^2-2bc(\cos A\cos\frac\pi3 - \sin A\sin\frac\pi3)\\ &= b^2+c^2-bc\cos A +\sqrt{3} bc\sin A\\ &= \frac{a^2+b^2+c^2}2 +\frac{\sqrt 3abc}{2R}, \end{aligned}$$ where $R$ is the radius of the circumcircle of $\triangle ABC$.
Then we can use, for example Heron's formula, to obtain a formula in $a$, $b$, and $c$ only.
Best Answer
Not a purely geometric construction, but an efficient algorithm (due to Weiszfeld) is well-known in the literature. These notes by Nam are an excellent survey on the topic, in my opinion.