<P> The converse also holds: If the sum of the distances from an interior point of a triangle to the sides is independent of the location of the point, the triangle is equilateral . </P> <P> Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plots, such as flammability diagrams . </P> <P> More generally, they allow one to give coordinates on a regular simplex in the same way . </P> <P> The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point . The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram . </P>

Let abc be an equilateral triangle and p be any point in the interior