Viviani's Theorem

Viviani's Theorem states that for any point $\mathcal{P}$ inside an equilateral triangle $\mathcal{ABC}$ the sum of the lengths of perpendiculars drawn from P on the sides is equal to the length of the altitude.
 

Draw equilateral triangles from that point onto the sides of the triangle such that these 3 perpendiculars are the altitudes of the equialteral triangles.Now rotate the triangles as shown.Now it can be easily seen that the sum of the 3 perpendiculars=perpendicular of the  $\Delta ABC$
 

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