Marinada '26

[lang=hr] U konveksnom četverokutu $ABCD$, točku $P$ unutar $ABCD$ nazivamo nonšalantnom ako, (1) $P$ ne leži na $AC,BD$ (2) Neka $AP,BP,CP,DP$ sijeku stranice $ABCD$ u točkama $A', B', C', D'$, redom, tada $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Pronađi maksimalni mogući broj nonšalantnih točaka [/lang] [lang=en] In a convex quadrilateral $ABCD$, we call a point $P$ in its interior nonchalant if (1) $P$ doesn't lie on lines $AC,BD$ (2) If $AP,BP,CP,DP$ intersects the sides of $ABCD$ at points $A', B', C', D'$ respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximal number of nonchalant points. [/lang]