Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.
Source: 2010 Saudi Arabia Pre-TST 4.2
Tags: triangle inequality, geometry, areas
Let $a$ be a real number. 1) Prove that there is a triangle with side lengths $\sqrt{a^2-a + 1}$, $\sqrt{a^2+a + 1}$, and $\sqrt{4a^2 + 3}$. 2) Prove that the area of this triangle does not depend on $a$.