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Neka je $ABCD$ tetivni četverokut takav da je $|AD| = |BD|$ i neka je $M$ sjecište njegovih dijagonala. Nadalje, neka je $N$ drugo sjecište dijagonale $\overline{AC}$ s kružnicom koja prolazi točkama $B$, $M$ i središtem kružnice upisane trokutu $BCM$.

Dokaži da vrijedi $|AN| \cdot |NC| = |CD| \cdot |BN|$.

Neka su točke $M$ i $N$ redom dirališta upisane kružnice raznostraničnog trokuta $ABC$ sa stranicama $\overline{AB}$ i $\overline{CA}$, a točke $P$ i $Q$ redom dirališta pripisanih kružnica nasuprot vrhova $B$ i $C$ s pravcem $BC$. Dokaži da je četverokut $MNPQ$ tetivan ako i samo ako je trokut $ABC$ pravokutan s pravim kutom pri vrhu $A$.

Zadan je tetivni četverokut $ABCD$ takav da se tangente u točkama $B$ i $D$ na njegovu opisanu kružnicu $k$ sijeku na pravcu $AC$. Točke $E$ i $F$ leže na kružnici $k$ tako da su pravci $AC$, $DE$ i $BF$ paralelni. Neka je $M$ sjecište pravaca $BE$ i $DF$. Ako su $P$, $Q$ i $R$ nožišta visina trokuta $ABC$, dokaži da točke $P$, $Q$, $R$ i $M$ leže na istoj kružnici.

Dan je tetivni četverokut $ABCD$. Polupravci $AD$ i $BC$ sijeku se u točki $P$. U unutrašnjosti trokuta $DCP$ dana je točka $M$ takva da pravac $PM$ raspolavlja kut $\measuredangle CMD$. Pravac $CM$ siječe kružnicu opisanu trokutu $DMP$ ponovno u točki $Q$, a pravac $DM$ siječe kružnicu opisanu trokutu $CMP$ ponovno u točki $R$.

a) Dokaži da dužine $\overline{CQ}$ i $\overline{DR}$ imaju jednaku duljinu.

b) Dokaži da trokuti $PAQ$ i $PBR$ imaju jednaku površinu.

Na stranici $\overline{AB}$ tetivnog četverokuta $ABCD$ postoji točka $X$ sa svojstvom da dijagonala $\overline{BD}$ raspolavlja dužinu $\overline{CX}$, a dijagonala $\overline{AC}$ raspolavlja dužinu $\overline{DX}$.

Koliki je najmanji mogući omjer $|AB| : |CD|$ u takvom četverokutu?

Dirališta upisane kružnice trokuta $ABC$ sa stranicama $\overline{AB}$ i $\overline{AC}$ su redom točke $D$ i $E$. Dirališta pripisane kružnice nasuprot vrha $A$ s pravcima $AB$ i $AC$ su redom točke $F$ i $G$.

Neka simetrale kutova $\measuredangle ABC$ i $\measuredangle ACB$ sijeku pravac $DE$ u točkama $X$ i $Y$ redom te neka vanjske simetrale kutova $\measuredangle ABC$ i $\measuredangle ACB$ sijeku pravac $FG$ u točkama $Z$ i $W$ redom.

Dokaži da je četverokut $XYZW$ tetivan.

Neka je $ABCD$ tetivan četverokut takav da je $|AB| = |AD|$. Točke $M$ i $N$ nalaze se redom na stranicama $\overline{BC}$ i $\overline{CD}$ i pritom je $|BM| + |DN| = |MN|$. Dokaži da središte opisane kružnice trokuta $AMN$ pripada pravcu $AC$.

Prove that if $n \geq 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.

In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$. Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.

$ABCD$ is cyclic. The feet of the perpendicular from $D$ to the lines $AB, BC, CA$ are $P, Q, R$ respectively. Show that the angle bisectors of $ABC$ and $CDA$ meet on the line $AC$ iff $RP = RQ$.

In a convex quadrilateral $ABCD$ the diagonal $BD$ does not bisect the angles $\angle ABC$ and $\angle CDA$. The point $P$ lies inside $ABCD$ and satisfies $$\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.$$

Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP$.

Consider five points $A, B, C, D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.

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

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD = AE$ and $CB = CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$, and $BC$ meet at one point.

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 $\angle AFS = \angle ECD$.

Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.

Neka je $ABC$ šiljastokutni trokut u kojem je $|AC| > |AB|$. Neka je $N$ nožište visine iz $A$ na stranicu $\overline{BC}$. Neka je točka $P$ na produžetku dužine $\overline{AB}$ preko vrha $B$, te neka je točka $Q$ na produžetku dužine $\overline{AC}$ preko vrha $C$ tako da je $BPQC$ tetivni četverokut. Ako vrijedi $|NP| = |NQ|$, dokaži da je $N$ središte kružnice opisane trokutu $APQ$.

Neka je $H$ ortocentar šiljastokutnog trokuta $ABC$. Kružnica opisana trokutu $ABH$ ima središte $S$ i siječe dužinu $\overline{BC}$ u točkama $B$ i $D$. Neka je $P$ presjek pravca $DH$ i dužine $\overline{AC}$, te neka je $Q$ središte opisane kružnice trokuta $ADP$. Dokaži da je četverokut $BDQS$ tetivan.