Marinada '24

[lang=hr] Neka je $ABC$ raznostranični trokut. Njemu upisana kružnica dodiruje stranice $BC$, $AC$ i $AB$ u točkama $A_1$, $B_1$ i $C_1$ redom. Pripisana kružnica nasuprot vrha $A$ dodiruje pravce $BC$, $AC$ i $AB$ u točkama $A_2$, $B_2$ i $C_2$, respektivno. Pretpostavimo da se pravci $AA_1$, $BB_1$, and $CC_1$ sijeku u točki $G$ i da se pravci $AA_2$, $BB_2$, and $CC_2$ sijeku u točki $G'$.Pravac $GG'$ siječe unutarnju simetralu kuta $\angle BAC$ u točki $T$. Pretpostavimo da je $AT=1$, $\cos{\angle BAC}=\sqrt{3}-1$ i da je $BC=8\sqrt[4]{3}$. Nađi $AB \cdot AC$. Odgovor zaokruži na $4$ decimale. [/lang] [lang=en] Let $ABC$ be a scalene triangle. The circle inscribed in it touches the sides $BC$, $AC$ and $AB$ at the points $A_1$, $B_1$ and $C_1$ respectively. The escribed circle opposite to vertex $A$ touches lines $BC$, $AC$, and $AB$ at points $A_2$, $B_2$, and $C_2$, respectively. Assume that lines $AA_1$, $BB_1$, and $CC_1$ intersect at point $G$ and that lines $AA_2$, $BB_2$, and $CC_2$ intersect at point $G'$. Line $GG' $ intersects the interior bisector of angle $\angle BAC$ at point $T$. Assume that $AT=1$, $\cos{\angle BAC}=\sqrt{3}-1$ and that $BC=8\sqrt[4]{3}$. Find $AB \cdot AC$. Round the answer to 4 decimal places. [/lang]