Math
Napoleon’s theorem
Erect an equilateral tile on each side of any triangle, mark the three centres and join them: an equilateral triangle, whatever the shape - then build the tiles inward, as a separate experiment, and compare the areas.
What happens
Take any triangle and build an equilateral triangle outward on each of its sides. Mark the centre of each and join the three centres: the result is always equilateral, and it is centred on the centroid of the original. Build the equilateral triangles inward instead and the three centres again make an equilateral triangle, smaller - and the two areas differ by exactly the area of the triangle you started from.
Equations used in the module
Equations on each side of ABC an equilateral triangle, erected outside (inside for the inner experiment) N₁, N₂, N₃ = the centres of the three equilateral triangles |N₁N₂| = |N₂N₃| = |N₃N₁| for every triangle ABC, flat ones included the centre of N₁N₂N₃ is the centroid of ABC area(outer) − area(inner) = area(ABC)
