Sum of Squares
$1+2^2+3^2+\cdots+n^2 = \frac{1}{3}n(n+1)\left(n+\frac{1}{2}\right)=\frac{n(n+1)(2n+1)}{6}$
Basel's Problem
$$1+\frac{1}{4}+\frac{1}{9}+\cdots = \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi ^2}{6}$$
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Sum of odd numbers
Proof of $$1+3+5+\cdots = n^{2}$$ Count the number of balls in the L shaped regions.Notice that they are odd numbers. Furthermore,count the number of balls considering the length or the breadth(they will turn out to be the same). Sum of all balls = (number of balls on the length)$^{2}$.
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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