4. Perpendiculars AP and AK are dropped from vertex A to the bisectors of the external angles ABC and ACB of triangle ABC. Find the length of segment PK, if the perimeter of triangle ABC is p.

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# Solution

![](https://cdn.mathpix.com/cropped/2024_05_06_4a4e2663a029be9bbe13g-2.jpg?height=748&width=1673&top_left_y=2079&top_left_x=223)

In triangle $\mathrm{ABC}$, we draw the external angle bisectors at vertices B and C. Drop perpendiculars from vertex A to these bisectors. Extend the constructed perpendiculars until they intersect the line containing side $\mathrm{BC}$ of triangle $\mathrm{ABC}$.

Triangle ABP is equal to triangle MBP, and triangle ACK is equal to triangle TCK. From this, we conclude that the corresponding sides of the triangles are equal and that segment PK is the midline of triangle AMT.

$$
\mathrm{PK}=\frac{\mathrm{MT}}{2}=\frac{\mathrm{MB}+\mathrm{BC}+\mathrm{CT}}{2}=\frac{\mathrm{AB}+\mathrm{BC}+\mathrm{CA}}{2}=\frac{\mathrm{p}}{2}
$$

## Evaluation Criteria

Consider individual steps that are essential for the solution - 1-3 points depending on progress in the problem.

The solution is completed, the answer is obtained, but there are inaccuracies in the justifications or reasoning - 4-6 points.

A complete and correctly justified solution to the problem - 7 points.