A point $D$ is chosen on side $B C$ of triangle $A B C$. The line tangent to the incircles of triangles $A B D$ and $A D C$ and different from $B C$ and $A D$ intersects segment $A D$ at $T$. If $A B=40 \text{ cm}, A C=50 \text{ cm}$, and $B C=60 \text{ cm}$, determine the value of the length of $A T$.

![](https://cdn.mathpix.com/cropped/2024_05_01_4a19aea8e08094ae376fg-07.jpg?height=539&width=1108&top_left_y=1079&top_left_x=546)
Solution

We know that the lengths of the tangent segments drawn from an external point to a circle are congruent and that $X Y=H I$. Thus,

$$
\begin{aligned}
2 A T & =(A F-T F)+(A J-T J) \\
& =A G+A K-(T X+T Y) \\
& =(A B-B G)+(A C-C K)-X Y \\
& =A B+A C-(B H+H I+I C) \\
& =A B+A C-B C \\
& =30
\end{aligned}
$$

Therefore, $A T=30 \text{ cm}$.