Školjka

%V0 U konveksnom četverokutu $ABCD$ vrijedi $|AD| = |CD|$ i $\angle{DAB} = \angle{ABC} < 90^\circ$. Pravac kroz $D$ i polovište $\overline{BC}$ siječe pravac $AB$ u točki $E$. Dokažite da je $\angle{BEC} = \angle{DAC}$.

%V0 Unutarnja točka $P$ u pravokutniku $ABCD$ odabrana je tako da je $\angle{BPC}+\angle{APD}=180^\circ$. Odredi sumu kutova $\angle{BCP}$ i $\angle{DAP}$.

%V0 Dan je tetivan četverokut $ABCD$, kojemu je $\overline{AC}$ promjer opisane kružnice. Neka točke $E$ i $F$ leže na pravcima $AD$ i $AB$. Dužine $\overline{EF}$ i $\overline{BD}$ se sijeku u $G$. Dokaži da ako je $AECF$ tetivan, da je $\angle EGC=90^\circ$.

%V0 Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$. Prove that $ABCDE$ has a pair of parallel sides.

%V0 Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$.

%V0 Suppose that $ABCD$ is a cyclic quadrilateral and $CD=DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\angle ADC=2\angle EDF$. Segments $DK$ and $DM$ are height and median of triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.