mathematics - Triangle with incircle

This question is from the German mathematics competition Känguru der Mathematik. In this competition students have to solve 30 mathematical tasks like this in 90 minutes without calculator. Actually they are given 5 possible answers, but for this community a bit of additional challenge does not hurt.

You are given a triangle $ABC$ with $\overline{AB} = 3$, $\overline{BC} = 5$ and $\overline{CA} = 6$ (image not to scale). The line segment $DE$ is tangential to the incircle of the triangle $ABC$. What is the perimeter of the triangle $CDE$?

Answer

This is not a puzzle but I want to share the answer:

First, we know that

The $|AB|$ is equal to $3$.

So

$|AF|+|BG|$ has to be equal to $3$ as well since the tangents to the circle from a point has to be equal to each other.

Moreover, E and D point location does not matter but the sum

$|EF|+|DG|$ is equal to $|DE|$ with the same rule. which is a part of the circumference of the big triangle except the grey part $|EC|+|DC|$.

As a result, whatever E and D point you choose, the circumference of $CDE$ will be

Circumference of CDE = Circumference of ABC - 6 = $6+5+3-6=8$