[Inscribed, circumscribed, and exscribed circles; their radii] [Pythagorean Theorem (direct and inverse).]

The sides of a triangle are in the ratio $5: 4: 3$. Find the ratios of the segments of the sides into which they are divided by the points of tangency with the inscribed circle.
The given triangle is a right triangle.

## Solution

As is known, the given triangle is a right triangle. We can assume that its sides are 3, 4, and 5. Then the radius of the inscribed circle is $1 / 2(3+4-5)=1$ (see problem 56847$)$.

The quadrilateral, whose vertices are the center of the circle, the points of tangency with the legs, and the vertex of the right angle of the triangle, is a square. Therefore, the distances from the vertices of the acute angles of the triangle to the points of tangency are 2 and 3.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-06.jpg?height=512&width=414&top_left_y=2225&top_left_x=822)

Problem