Four Consecutive Numbers

Find a triangle whose three sides and one of its heights are four consecutive whole numbers.

Solution

Take the triangle with sides $13$, $14$, $15$, and drop the height onto the side of length $14$. It meets that side at the point splitting it into lengths $5$ and $9$. Since $$5^2 + 12^2 = 13^2 \qquad \text{and} \qquad 9^2 + 12^2 = 15^2,$$ the two pieces are the bases of right triangles of height $12$ whose slanting sides are $13$ and $15$. Glued along their shared side of length $12$, they form a triangle with base $5 + 9 = 14$, other sides $13$ and $15$, and height $12$. So the three sides and the height are $12, 13, 14, 15$, four consecutive whole numbers.

(Equivalently, Heron’s formula gives the area of the $13, 14, 15$ triangle as $84$, so the height onto the side of length $14$ is $2 \times 84 / 14 = 12$.)