Viviani's theorem

Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.

[1] It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.

Let P be any point inside the triangle, and s, t, u the perpendicular distances of P from the sides.

Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA.

[3] Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plots, such as flammability diagrams.

The converse in general is not true, as the result holds for an equilateral hexagon, which does not necessarily have opposite sides parallel.

[1] A necessary and sufficient condition for a convex polygon to have a constant sum of distances from any interior point to the sides is that there exist three non-collinear interior points with equal sums of distances.

For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.
Visual proof of Viviani's theorem
1. Shortest distances from point P to sides of equilateral triangle ABC are shown.
2. Lines DE, FG, and HI parallel to AB, BC and CA, respectively, and passing through P define similar triangles PHE, PFI and PDG.
3. As these triangles are equilateral, their altitudes can be rotated to be vertical.
4. As PGCH is a parallelogram, triangle PHE can be slid up to show that the altitudes sum to that of triangle ABC.
Flammability diagram for methane