227. Find the triangle. The sides and height of a certain triangle are expressed by four consecutive integers. What is the area of this triangle?
227. The sides of a triangle are 13, 14, and 15, with the base being 14, the height 12, and the area 84. There are infinitely many rational triangles whose sides are expressed as consecutive integers, such as 3, 4, and 5 or 13, 14, and 15, but only one of them satisfies our conditions for the height.

Triangles with sides expressed as three consecutive integers and an integer area are as follows:

\[
\begin{array}{lll}
3 & 4 & 5
\end{array}
\]

\(\begin{array}{lll}13 & 14 & 15\end{array}\)

\(\begin{array}{lll}51 & 52 & 53\end{array}\)

193 194 195

723 724 725 and so on.

They can be found very easily:

\[
\begin{aligned}
52 & =4 \times 14-4 \\
194 & =4 \times 52-14 \\
724 & =4 \times 194-52
\end{aligned}
\]

or in general \(U_{n}=4 U_{n-1}-U_{n-2}\). There is another way to construct these triangles. Find \(x\) such that \(3\left(x^{2}-1\right)\) is a perfect square. It will correspond to a triangle with sides \(2 x, 2 x+1, 2 x-1\).