Marinada '22

[lang=hr] U $\triangle ABC$, $|CA| = 1960\sqrt{2}$ , $|CB| = 6720$ i $\angle ACB = 45^o$. Neka su $K$, $L$ , $M$ točke koje leže na $\overline{BC}$, $\overline{CA}$ i $\overline{AB}$ takve da $\overline{AK} \perp \overline{BC}$ , $\overline{BL} \perp \overline{CA}$, $|AM| = |BM|$. Neka su $N$, $O$, $P$ točke koje leže na $\overline{KL}$ , $\overline{BA}$ i $\overline{BL}$ t.d. je $|AN| = |KN|$ , $|BO| = |CO|$ i $A$ leži na $\overline{NP}$. Ako je $H$ ortocentar od $\triangle MOP$, izračunaj $|HK|^2$. [/lang] [lang=en] In $\triangle ABC$, $|CA| = 1960\sqrt{2}$ , $|CB| = 6720$ and $\angle ACB = 45^o$. Let $K$, $L$ , $M$ be points that lie on $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ such that $\overline{AK} \perp \overline{BC}$ , $\overline{BL} \perp \overline{CA}$, $|AM| = |BM|$. Let $N$, $O$, $P$ be points that lie on $\overline{KL}$ , $\overline{BA}$ and $\overline{BL}$ such that $|AN| = |KN|$ , $|BO| = |CO|$ and $A$ lies on $\overline{NP}$. If $H$ is the orthocenter of $\triangle MOP$, determine $|HK|^2$. [/lang]