Marinada '22

[lang=hr] Zadan je trokut $ABC$ takav da vrijedi $|AB| = \frac{\overline{AC}}{2} + \overline{BC}$. Izvan trokuta konstruirana su dva polukruga kojima su promjeri $\overline{AB}$ i $\overline{BC}$. Neka je $T$ nožište iz $A$ na zajedničku tangentu polukrugova. Odredi $\angle CAT$. (rješenje zapišite u stupnjevima bez mjerne jedinice npr. "45") [/lang] [lang=en] Consider a triangle $ABC$ in which $|AB| = \frac{\overline{AC}}{2} + \overline{BC}$. Outside of that triangle we construct two semicircles with diameters $\overline{AB}$ and $\overline{BC}$. Let $T$ be the point at intersection of perpendicular from $A$ with the mutual tangent of semicircles. Determine $\angle CAT$. (Write the solution in degrees but without measuring unit, e.g. "45".) [/lang]