Let $a, b, c$ be the side lengths of the triangle, and $x, y, z$ the distances from a point inside a triangle to the respective vertices. Then the numbers $x,y,z$ satisfy the equation
$$\begin{vmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & x^2 & y^2 & z^2\\
1 & x^2 & 0 & c^2 & b^2\\
1 & y^2 & c^2 & 0 & a^2\\
1 & z^2 & b^2 & a^2 & 0
\end{vmatrix} = 0$$
This is the [Cayley-Menger determinant][1],
for pairwise distances between four points in $\mathbb{R}^3$ it is proportional to the square of the volume of the tetrahedron they span. The volume vanishes if and only if four points are coplanar.


  [1]: https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant