The sides of a triangle are a, b and c where a, b and c are integers and $\displaystyle a\leq b\leq c$. If c is given, show that the number of different triangles is $\displaystyle \frac{1}{4}c(c + 2)$ or $\displaystyle \frac{1}{4}(c + 1)^2$, according as c is even or odd. Also, show that the number of isoceles or equilateral triangle is $\displaystyle \frac{1}{2}(3c - 2)$ or $\displaystyle \frac{1}{2}(3c - 1)$, according as $\displaystyle c$ is even or odd.