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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, 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.