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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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 $\Delta ABC$
Fibonacci Sum of Squares
One can easily see that the length and breadth of the above rectangle are $\mathcal{F_{n+1}}$and $\mathcal{F_{n}}$. Area of the rectangle = $\sum$ Area of the squares or,$$\mathcal{F_{n}F_{n+1}=F_{0}^{2}+F_{1}^{2}+\cdots+F_{n}^{2}}$$
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