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U šiljastokutnom trokutu $ABC$, u kojem je $|AC| < |BC|$, točke $M$ i $N$ su redom nožišta visina iz vrhova $A$ i $B$. Kružnica sa središtem $O$ opisana trokutu $ABC$ i kružnica sa središtem $S$ opisana trokutu $MNC$ sijeku se u točkama $C$ i $D$. Ako je točka $P$ polovište dužine $\overline{AB}$, dokaži da točke $P, O, S$ i $D$ leže na istoj kružnici.

Neka je $ABC$ šiljastokutni trokut u kojem je $|AC| > |BC|$. Neka je $H$ ortocentar tog trokuta, $N$ nožište visine iz vrha $B$, a $P$ polovište dužine $\overline{AB}$. Kružnice opisane trokutima $ABC$ i $CHN$ sijeku se u točkama $C$ i $D$. Dokaži da točke $B$, $D$, $N$ i $P$ leže na istoj kružnici.

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 jednakokračan trokut $ABC$ takav da je $|AB| = |AC|$. Neka je $M$ polovište stranice $\overline{BC}$ te neka je $P$ točka različita od $A$ takva da je $PA \parallel BC$. Točke $X$ i $Y$ nalaze se redom na polupravcima $PB$ i $PC$, tako da je točka $B$ između $P$ i $X$, točka $C$ između $P$ i $Y$ te vrijedi $\measuredangle PXM = \measuredangle PYM$. Dokaži da su točke $A$, $P$, $X$ i $Y$ konciklične.

Neka je $T$ točka unutar šiljastokutnog trokuta $ABC$ i neka su $A_1$, $B_1$ i $C_1$ točke osnosimetrične točki $T$ u odnosu na pravce $BC$, $CA$ i $AB$, redom. Pravci $A_1T$, $B_1T$ i $C_1T$ sijeku kružnicu $k$ opisanu trokutu $A_1B_1C_1$ ponovno u točkama $A_2$, $B_2$ i $C_2$, redom.

Dokaži da se pravci $AA_2$, $BB_2$, $CC_2$ sijeku u jednoj točki koja leži na kružnici $k$.

Dan je konveksan četverokut $ABCD$ čije se dijagonale sijeku u točki $P$. Neka su $X$ i $Y$ točke odabrane tako da četverokuti $ABPX$, $CDXP$, $BCPY$ i $DAYP$ budu tetivni. Pravci $AB$ i $CD$ sijeku se u točki $Q$, pravci $BC$ i $DA$ u točki $R$, a pravci $XR$ i $YQ$ u točki $Z$. Dokaži da točke $X$, $Y$, $Z$ i $P$ pripadaju istoj kružnici.

Let $A, B, C, D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM, DN, XY$ are concurrent.

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.

Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.

An acute-angled triangle $ABC$ has orthocentre $H$. The circle passing through $H$ with centre the midpoint of $BC$ intersects the line $BC$ at $A_1$ and $A_2$. Similarly, the circle passing through $H$ with centre the midpoint of $CA$ intersects the line $CA$ at $B_1$ and $B_2$, and the circle passing through $H$ with centre the midpoint of $AB$ intersects the line $AB$ at $C_1$ and $C_2$. Show that $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a circle.

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C = \angle BAC$. Similarly, let $Q_1$ be a point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q = \angle CBA$.

Prove that points $P$, $Q$, $P_1$, and $Q_1$ are concyclic.

Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.

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.

Let $ABC$ be an acute triangle with $AB > AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and $$\angle AXB - \angle ACB = \angle CYA - \angle CBA$$ holds, the line $XY$ passes through $D$.

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let $D$ be the foot of its altitude from $A$. Let $R$ and $Q$ be the centroids of the triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and the points $P, Q, R$ and $D$ are concyclic. Prove that the lines $AP, BQ$ and $CR$ are concurrent.

Let $ABC$ be an acute-angled triangle with $AC > BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP = AC$ and that $P$ is an interior point of the shorter arc $BC$ of $\omega$. Let $Q$ be the point of intersection of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA = QR$ and that $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P$, $Q$, $R$, and $S$ are concyclic.

Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A$, $D$, $E$ and $F$ are concyclic.

We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove that the points $P, Q, R, S$ are concyclic.

Dan je četverokut $ABCD$. Opisana kružnica trokuta $ABC$ siječe stranice $\overline{CD}$ i $\overline{DA}$ redom u točkama $P$ i $Q$, a opisana kružnica trokuta $CDA$ stranice $\overline{AB}$ i $\overline{BC}$ redom u točkama $R$ i $S$. Pravci $BP$ i $BQ$ sijeku pravac $RS$ redom u točkama $M$ i $N$. Dokaži da točke $M$, $N$, $P$ i $Q$ leže na istoj kružnici.

Dan je trapez $ABCD$ kojem su kutovi uz osnovicu $\overline{AB}$ šiljasti, a dijagonale su mu međusobno okomite i sijeku se u točki $O$. Polupravac $OA$ siječe kružnicu s promjerom $\overline{BD}$ u točki $M$, a polupravac $OB$ siječe kružnicu s promjerom $\overline{AC}$ u točki $N$.

Dokaži da točke $M$, $N$, $C$ i $D$ leže na jednoj kružnici.

Dan je šiljastokutan trokut $ABC$ u kojem vrijedi $|AC| > |AB|$, a točka $O$ je središte opisane kružnice. Simetrala kuta $\measuredangle CAB$ siječe stranicu $\overline{BC}$ u točki $D$. Pravac okomit na pravac $AD$ koji prolazi kroz točku $B$ siječe pravac $AO$ u točki $E$.

Dokaži da točke $A$, $B$, $D$ i $E$ leže na istoj kružnici.

Na stranici $\overline{AC}$ trokuta $ABC$ nalaze se točke $D$ i $E$ tako da je točka $D$ između $C$ i $E$. Neka je $F$ sjecište kružnice opisane trokutu $ABD$ s pravcem koji prolazi kroz točku $E$ i paralelan je s $BC$ tako da se točke $E$ i $F$ nalaze s različitih strana pravca $AB$. Neka je $G$ sjecište kružnice opisane trokutu $BCD$ s pravcem koji prolazi kroz točku $E$ i paralelan je s $AB$ tako da se točke $E$ i $G$ nalaze s različitih strana pravca $BC$.

Dokaži da točke $D$, $E$, $F$ i $G$ leže na istoj kružnici.

Dan je šiljastokutni trokut $ABC$ u kojem vrijedi $|AB| > |AC|$. Neka je $O$ središte kružnice opisane tom trokutu, a $\overline{OQ}$ promjer kružnice opisane trokutu $BOC$. Pravac paralelan s pravcem $BC$ kroz $A$ siječe pravac $CQ$ u točki $M$, a pravac paralelan s pravcem $CQ$ kroz $A$ siječe pravac $BC$ u točki $N$. Neka je $T$ presjek pravaca $AQ$ i $MN$.

Dokaži da točka $T$ leži na kružnici opisanoj trokutu $BOC$.