[Pythagorean Theorem (direct and inverse).] [ Rhombi. Properties and criteria ]

A circle with a radius of 3 is inscribed in a triangle. Find the sides of the triangle if one of them is divided by the point of tangency into segments equal to 4 and 3.
The given triangle is a right triangle.

## Solution

Let the incircle touch the sides $A B, A C, B C$ of triangle $A B C$ at points $K, M$ and $N$ respectively, with $A K=3, K B=4$.

If $O$ is the center of the circle, then $A M O K$ is a rhombus with two right angles, which means it is a square. Therefore, $\angle A=90^{\circ}$. Let $C M=C N=x$. By the Pythagorean theorem,

$(3+x)^{2}+7^{2}=(4+x)^{2}$, from which $x=21$. Therefore, $A C=C M+A M=24, B C=B N+C N=25$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-33.jpg?height=343&width=766&top_left_y=0&top_left_x=649)

## Answer

$7,25,24$.