243. Two medians of a triangle are perpendicular to each other. Find the sides of the triangle, knowing that they are expressed in whole numbers.
243. Denoting the sides corresponding to the medians specified in the condition by \(x\) and \(y\), and the third side by \(z\), and using the property of the intersection point of the medians, we obtain the equation \(x^{2} + y^{2} = 5 z^{2} (1)\).

To solve equation (1), rewrite it as \(x^{2} - 4 z^{2} = z^{2} - y^{2}\), or \(\frac{x + 2z}{z + y} = \frac{z - y}{x - 2z} = \frac{m}{n}\), which gives the solutions \(x = 2(m^{2} + mn - n^{2}), y = n^{2} + 4mn - m^{2}, z = m^{2} + n^{2}\). These formulas define the sides of the triangle. For definiteness, assume \(x > y\). Solve the systems of inequalities for \(x > 0, y > 0, z > 0, x > y, z > x - y\), we get \(\frac{\sqrt{10} + 1}{3} < \frac{m}{n} < 2\).

For example, let \(m = 3, n = 2\), then \(x = 22, y = 19\), \(z = 13\).