Triangle whose sides and altitude are consecutive

A triangle has three side lengths and one altitude that, taken together, form four consecutive positive integers. Find its area.

84. The $13$–$14$–$15$ triangle works: by Heron's formula its area is $\sqrt{21\cdot 8\cdot 7\cdot 6}=84$, so the altitude onto the side of length $14$ is $2\cdot 84/14=12$, and $12,13,14,15$ are consecutive.