Marinada '25

[lang=hr] Trokut $ABC$ upisan je u kružnicu $\omega$ pri čemu $AB = 5$, $BC = 7$ i $AC = 3$. Simetrala unutarnjeg kuta $A$ siječe $BC$ u $D$ i kružnicu drugi put $\omega$ u točki $E$. Neka je $\gamma$ kružnica promjera $DE$. Kružnice $\omega$ i $\gamma$ sijeku se u $E$ i $F$. Imamo $AF^2 = \frac mn$, gdje m i n su relativno prosti prirodni brojevi. Nađi $m + n$. [/lang] [lang=en] In the triangle $ABC$, which is inscribed in the circle $\omega$, it is known that $AB = 5$, $BC = 7$ and $AC = 3$. The two points $D$ and $E$ are the points where the bisector of angle $A$ intersects side $BC$ and circle $\omega$, respectively. Let $\gamma$ be the circle with a diameter $DE$. The circles $\omega$ and $\gamma$ intersect at points $E$ and $F$. Additionally, $AF^2 = \frac mn$, where $m$ and $n$ are relatively prime natural numbers. What is the value of $m+n$? [/lang]