I am trying to solve the above question, as an application of Sylow's theorem.
Let $P$ be the p-Sylow subgroup. Then $n_p | (p+1)$ and $n_p \equiv 1 \pmod{p}$. If $n_p =1$, $P$ is normal and we are done, else $n_p = p+1$. Now,
\begin{equation}
1+n_p(p-1) = 1 + (p+1)(p-1) = p^2,
\end{equation}
...
In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the centres of those equilateral triangles themselves form an equilateral triangle.
The triangle thus formed is called the inner or outer Napoleon triangle. The difference in area of these two triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may date back to W. Rutherford's 1825 question published in The Ladies' Diary, four years after the French emperor...
