Školjka

%V0 Kroz zadanu točku $P$ na nekoj stranici trokuta $ABC$ konsturiraj pravac koji će trokut podijeliti na dva dijela jednake površine.

%V0 Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that $$AB+AC\geq4DE.$$

%V0 Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

%V0 Let ABC be a triangle and $M$ be an interior point. Prove that $$\min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.$$

%V0 Let $M$ and $N$ be two points inside triangle $ABC$ such that $$\angle MAB = \angle NAC\quad \mbox{and}\quad \angle MBA = \angle NBC.$$ Prove that $$\frac {AM \cdot AN}{AB \cdot AC} + \frac {BM \cdot BN}{BA \cdot BC} + \frac {CM \cdot CN}{CA \cdot CB} = 1.$$

%V0 Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.