Školjka

%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 Let $ABCD$ be a convex quadrilateral such that $AB$ and $CD$ are not parallel and $AB=CD$. The midpoints of the diagonals $AC$ and $BD$ are $E$ and $F$, respectively. The line $EF$ meets segments $AB$ and $CD$ at $G$ and $H$, respectively. Show that $\angle AGH = \angle DHG$.

%V0 Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$. Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$.

%V0 The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC \perp BD.$ Four squares, $ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $CL, DF, AH, BJ$ are denoted by $P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $AI, BK, CE DG$ are denoted by $P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $P_1,Q_1,R_1, S_1$ and $P_2,Q_2,R_2, S_2$ are the two quadrilaterals. Alternative formulation: Outside a convex quadrilateral $ABCD$ with perpendicular diagonals, four squares $AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $Q_1$ and $Q_2$ formed by the lines $AG, BI, CK, DE$ and $AJ, BL, CF, DH,$ respectively, are congruent.

%V0 $ABCD$ is a terahedron: $AD+BD=AC+BC,$ $BD+CD=BA+CA,$ $CD+AD=CB+AB,$ $M,N,P$ are the mid points of $BC,CA,AB.$ $OA=OB=OC=OD.$ Prove that $\angle MOP = \angle NOP =\angle NOM.$

%V0 Let $R$ be a rectangle that is the union of a finite number of rectangles $R_i,$ $1 \leq i \leq n,$ satisfying the following conditions: (i) The sides of every rectangle $R_i$ are parallel to the sides of $R.$ (ii) The interiors of any two different rectangles $R_i$ are disjoint. (iii) Each rectangle $R_i$ has at least one side of integral length. Prove that $R$ has at least one side of integral length. Variant: Same problem but with rectangular parallelepipeds having at least one integral side.