Marinada '23

[lang=hr] U trokutu $ABC$, $AB=3$, $AC=5$ i $BC=7$. Neka je $E$ osnosimetrična točka točki $A$ s obzirom na pravac $BC$ i neka je pravac $BE$ siječe trokutu $ABC$ opisanu kružnicu u točki $D$, $D\neq B$. Neka je $I$ središte trokutu $ABD$ upisane kružnice. Odredi $\cos {\angle AEI}$. Odgovor zaokružite na tri decimale. [/lang] [lang=en] In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over line $BC$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Find $\cos {\angle AEI}$. Round your answer to $3$ decimal places. [/lang]