Marinada '26

[lang=hr] Neka je $ABC$ šiljastokutni trokut gdje su $\omega, S$, njegova upisana kružnica i opisana kružnica, redom. Kružnica $\omega'$ je unutarnje tangetna kružnici $S$ u $A$ te vanjsko tangentna kružnici $\omega$. Kružnica $S'$ je unutarnje tangentna kružnici $S$ u $A$ te unutarnje tangentna kružnici $\omega$. Neka su $P'$ i $Q'$ centri $\omega'$ i $S'$, redom. Izračunajte duljinu dužine $P'Q'$ ako su duljine stranica trokuta $a$, $b$, $c$ jednake 4, 13 i 15, redom [/lang] [lang=en] Let $ABC$ be an acute angled triangle with incircle $\omega$ and circumcircle $S$. Assume that circle $\omega'$ is internally tangent to $S$ in $A$ and externally tangent to $\omega$. Circle $S'$ is internally tangent to $S$ at $A$ and internally tangent to $\omega$. Let $P'$ and $Q'$ be centers of $\omega'$ and $S'$ respectively. Find the length of $P'Q'$ given that the side lengths of the triangle $a$, $b$, $c$ are equal to 4, 13 and 15 respectively. [/lang]