Marinada '23

[lang=hr] Dan je raznostranični trokut $ABC$ s opisanom kružnicom $k$ i opsegom $300$. Točke $K,L$ su središta lukova $ABC$ i $ACB$ kružnice $k$. Tangenta u $A$ na kružnicu $k$ siječe polupravac $KL$ u $M$. Poznato je da polovište dužine $AM$ leži na $BC$. Odredi duljinu $BC$. [/lang] [lang=en] A scalene triangle $ABC$ with circumscribed circle $k$ and perimeter $300$ is given. The points $K,L$ are the centers of the arcs $ABC$ and $ACB$ of the circle $k$. The tangent at $A$ to the circle $k$ intersects the ray $KL$ at $M$. It is known that the midpoint of $AM$ lies on $BC$. Determine the length of $BC$. [/lang]