Proof
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, and . 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:
and thus
- u + s + t = h.
Q.E.D.
Read more about this topic: Viviani's Theorem
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