5. In an isosceles triangle $M N K$, the sides $M N = N K = 12$, and $M K = 8$. A point $F$ is chosen on side $N K$ such that the circles inscribed in triangles $M N F$ and $M K F$ touch each other. Find the areas of triangles $M N F$ and $M K F$.
Answer: $S_{M N F}=\frac{32 \sqrt{2}}{3}, S_{M K F}=\frac{64 \sqrt{2}}{3}$.

## Solution.

Let $R, L, E, P, Q$ be the points of tangency, and $x, y, u, v$ be the lengths of the tangent segments as indicated in the diagram.

Then

$$
\left\{\begin{array} { c } 
{ x + y = 1 2 , } \\
{ y + u = 8 } \\
{ x + 2 v + u = 1 2 . }
\end{array} \Rightarrow \left\{\begin{array}{r}
x-u=(x+v)-(u+v)=4 \\
(x+v)+(u+v)=12
\end{array}\right.\right.
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_67fb85ec2e8269040458g-4.jpg?height=523&width=348&top_left_y=1235&top_left_x=1388)

Therefore,

$$
\begin{aligned}
& \frac{S_{M N F}}{S_{M K F}}=\frac{N F}{F K}=2 \Rightarrow S_{M K F}=\frac{1}{3} S_{M N K}=\frac{1}{6} M K \sqrt{M N^{2}-\frac{(M K)^{2}}{4}}=\frac{8}{6} \sqrt{12^{2}-\frac{8^{2}}{4}}=\frac{4}{3} \sqrt{128}=\frac{32 \sqrt{2}}{3} . \\
& \quad \text { Then } S_{M N F}=2 S_{M K F}=\frac{64 \sqrt{2}}{3} .
\end{aligned}
$$

## Evaluation criteria are provided in the table:

| Points | $\quad$ Evaluation Criteria |
| :---: | :--- |
| $\mathbf{7}$ | Fully justified solution. |
| $\mathbf{6}$ | Justified solution with minor flaws. |
| $\mathbf{5 - 6}$ | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct with minor corrections or additions. |
| $\mathbf{4}$ | The problem is more solved than not, for example, one of the two (more complex) significant cases is correctly considered. |
| $\mathbf{2 - 3}$ | The problem is not solved, but formulas, diagrams, considerations, or auxiliary statements relevant to the solution of the problem are provided. |
| $\mathbf{1}$ | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or in the case of an incorrect solution. |
| $\mathbf{0}$ | The solution is absent, or the solution does not meet any of the criteria listed above. |