Gubin Ya.

The lengths of the sides of a certain triangle and the diameter of the circle inscribed in it are four consecutive terms of an arithmetic progression. Find all such triangles.
All triangles with side lengths proportional to 3, 4, and 5 (Egyptian triangle). In any triangle, \(2 r < a < 2 r + 2 d\). Clearly, \(a = 2 r + d, b = 2 r + 2 d\),

![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-08.jpg?height=75&width=1889&top_left_y=1529&top_left_x=16)
\(\sqrt{p(p-a)(p-b)(p-c)}\) or \(p r^{2} = (p-a)(p-b)(p-c)\). Expressing all quantities in this equation in terms of \(r\) and \(d\), we get

\[
(3 r + 3 d) r^{2} = (r + 2 d)(r + d) r
\]

from which \(3 r = r + 2 d\), i.e., \(r = d\), since \(r > 0, r + d > 0\). Therefore, the sides are \(3 r, 4 r, 5 r\).

## Answer

All triangles with side lengths proportional to 3, 4, and 5 (Egyptian triangle).