Heights and a Hidden Identity

In $\triangle ABC$, the altitudes to sides $a, b, c$ have lengths $h_a, h_b, h_c$, and

1. If $S$ is the area of the triangle, prove that $S = \dfrac{1}{12}\left(3a^2 - b^2 + 6c^2\right)$. 2. Express $\sin\left(A + \dfrac{\pi}{4}\right)$ in terms of $b$ and $c$, and find $A$. 3. Based on the structure of this problem, formulate an analogous statement about triangles of your own design, and prove it.