[Linear element relationships of similar triangles] Signs and properties of the tangent

A semicircle is inscribed in a triangle, with the semicircle touching the base, and the diameter (with endpoints on the lateral sides of the triangle) parallel to the base. Find the radius of the semicircle if the base of the triangle is $a$, and the height is $h$.
The heights of similar triangles are proportional to the corresponding sides.

## Solution

Let $M$ and $N$ be the endpoints of the diameter of the semicircle, lying on the sides $A B$ and $A C$ of triangle $A B C$ with base $B C=a$ and height $h$, and $R$ be the radius of the semicircle.

Since triangles $A M N$ and $A B C$ are similar, we have $\quad{ }^{h-R} / 2_{R}=h / a$. From this, $R=a h / a+2 h$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-20.jpg?height=297&width=604&top_left_y=1944&top_left_x=727)

Answer

$a h_{/ a+2 h}$.