6. Given triangle $A B C$. Lines $O_{1} O_{2}, O_{1} O_{3}, O_{3} O_{2}$ are the bisectors of the external angles of triangle $A B C$, as shown in the figure. Point $O$ is the center of the inscribed circle of triangle $A B C$. Find the angle in degrees between the lines $O_{1} O_{2}$ and $O O_{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b77a4ddf94595f81d054g-02.jpg?height=508&width=685&top_left_y=2025&top_left_x=731)
Solution. Point $O$ is the intersection point of the angle bisectors of triangle $ABC$, therefore, the bisector $BO$ is perpendicular to the line $O_{1} O_{2}$ (as the bisectors of adjacent angles of the triangle).

![](https://cdn.mathpix.com/cropped/2024_05_06_b77a4ddf94595f81d054g-03.jpg?height=546&width=717&top_left_y=178&top_left_x=721)

Point $O_{3}$, being equidistant from the lines $BA$ and $BC$, lies on $BO$. Therefore, the line $OO_{3}$, which coincides with $BO$, is perpendicular to the line $O_{1} O_{2}$.

Answer: 90 .