Školjka

%V0 U trokutu $ABC$ dane su točke $D, E, F$ na stranicama $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ tako da je $|AF|=\frac{1}{n}|AB|$, $|BD|=\frac{1}{n}|BC|$, $|CE|=\frac{1}{n}|CA|$. Pravci $AD, BE, CF$ sijeku se u točkama $G, H, I$. Pokažite da je površina trokuta $GHI$ jednaka $$ P_{GHI} = \frac{(n-2)^2}{n^2-n+1} P_{ABC} .$$

%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 $ABCD$ 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 $F$.

%V0 Neka je $ABC$ trokut u kojem vrijedi $\left\vert AC \right\vert > \left\vert BC \right\vert$. Izrazi površinu trokuta određenog stranicom $\overline{AB}$, simetralom stranice $\overline{AB}$ i simetralom kuta $\angle{ACB}$ pomoću duljina stranica trokuta $ABC$.