Viviani's Theorem

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

This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.

Let ABC be an equilateral triangle whose height is h and whose side is a.

Let P be any point inside the triangle, and u, s, t the 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.

Now, the areas of these triangles are ${\displaystyle {\frac {u\cdot a}{2}}}$, ${\displaystyle {\frac {s\cdot a}{2}}}$, and ${\displaystyle {\frac {t\cdot a}{2}}}$. They exactly fill the enclosing triangle, so the sum of these areas is equal to the area of the enclosing triangle. So we can write:

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