Finding Sides of a Triangle

Triangle problems often involve finding minimum/maximum side lengths based on whether the triangle is right, acute, or obtuse. Key formulas are:

• Triangle Inequality: $a+b > c$
• Right Triangle: $a^2 + b^2 = c^2$
• Acute Triangle: $a^2 + b^2 > c^2$
• Obtuse Triangle: $a^2 + b^2 < c^2$ (where c is the longest side)

A trick for finding integer sides of a right triangle given one leg, $a$: If $a$ is odd, the other sides are $\frac{a^2-1}{2}$ and $\frac{a^2+1}{2}$. For example, if $a=9$, $a^2=81$, the other sides are 40 and 41.