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U četverokutu $ABCD$ je $\measuredangle DAB = 110^\circ$, $\measuredangle ABC = 50^\circ$, $\measuredangle BCD = 70^\circ$. Neka su $M$ i $N$ polovišta dužina $\overline{AB}$ i $\overline{CD}$ redom. Za točku $P$ na dužini $\overline{MN}$ vrijedi $|AM| : |CN| = |MP| : |NP|$ i $|AP| = |CP|$. Odredi veličinu $\measuredangle APC$.

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?

Neka je $ABCD$ konveksni četverokut u kojem je $\measuredangle B > 90°$, $\measuredangle D > 90°$ te $\measuredangle A = \measuredangle C$. Neka su $E$ i $F$ redom točke osnosimetrične točki $A$ u odnosu na pravce $BC$ i $CD$. Neka dužine $\overline{AE}$ i $\overline{AF}$ sijeku pravac $BD$ redom u točkama $K$ i $L$.

Dokaži da se kružnice opisane trokutima $BKE$ i $FLD$ diraju.

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.

Neka je $ABCD$ konveksni četverokut čije se dijagonale sijeku u točki $E$. Pravci $AB$ i $CD$ sijeku se u točki $P$, a pravci $AD$ i $BC$ u točki $Q$. Neka je $X$ sjecište kružnica opisanih trokutima $EBC$ i $EDA$ različito od $E$, a $Y$ sjecište kružnica opisanih trokutima $EAB$ i $ECD$ različito od $E$. Konačno, neka je $W$ sjecište kružnica opisanih trokutima $PBC$ i $PDA$ različito od $P$. Dokaži da su trokuti $WQY$ i $WXP$ slični.

Neka je $ABCD$ tetivni četverokut. Neka su $M$ i $N$ redom polovišta dužina $\overline{BC}$ i $\overline{AD}$. Pretpostavimo da točke $Q, A, B, P$ leže na pravcu u tom poretku, da je $AC$ tangenta opisane kružnice trokuta $ADQ$ te da je $BD$ tangenta opisane kružnice trokuta $BCP$. Dokaži da se pravac $CD$, tangenta opisane kružnice trokuta $ANQ$ u točki $A$ i tangenta opisane kružnice trokuta $BMP$ u točki $B$ sijeku u jednoj točki.

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 $AC$.

On the circle $K$ there are given three distinct points $A,B,C.$ Construct (using only straightedge and compasses) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Given $n > 4$ points in the plane such that no three are collinear. Prove that there are at least $\binom{n-3}{2}$ convex quadrilaterals whose vertices are four of the given points.

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.

In a plane convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

Let $ABCD$ be a convex quadrilateral such that the line $CD$ is a tangent to the circle on $AB$ as diameter. Prove that the line $AB$ is a tangent to the circle on $CD$ as diameter if and only if the lines $BC$ and $AD$ are parallel.

Let $ABCD$ be a convex quadrilateral such that the sides $AB$, $AD$, $BC$ satisfy $AB = AD + BC$. There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP = h + AD$ and $BP = h + BC$. Show that:

$$\frac{1}{\sqrt{h}} \geq \frac{1}{\sqrt{AD}} + \frac{1}{\sqrt{BC}}.$$

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.

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 $AP = CP$.

Let $ABCD$ be a fixed convex quadrilateral with $BC = DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE = DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$.

Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

Let $ABCD$ be a convex quadrilateral with $|BA| \neq |BC|$. Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_1$ and $\omega_2$ respectively. Suppose that there exists a circle $\omega$ tangent to the ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$. Prove that the common external tangents of $\omega_1$ and $\omega_2$ intersect on $\omega$.

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90°$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and $$\angle CHS - \angle CSB = 90°, \quad \angle THC - \angle DTC = 90°.$$

Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

A convex quadrilateral $ABCD$ satisfies $AB \cdot CD = BC \cdot DA$. Point $X$ lies inside $ABCD$ so that $$\angle XAB = \angle XCD \quad \text{and} \quad \angle XBC = \angle XDA.$$ Prove that $\angle BXA + \angle DXC = 180°$.

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $$\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC.$$

Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.

Let $\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB$, $BC$, $CD$ and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $BC$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $AD$ and $CD$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that $$AD + DT + TX + XA = CD + DY + YZ + ZC.$$

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 $F$. The line $EF$ meets segments $AB$ and $CD$ at $G$ and $H$, respectively. Show that $\measuredangle AGH = \measuredangle DHG$.

Let $ABCD$ be a convex quadrilateral such that $AC = BD$ and the sides $AB$ and $CD$ are not parallel. Let $P$ be the intersection point of the diagonals $AC$ and $BD$. Points $E$ and $F$ lie, respectively, on segments $BP$ and $AP$ such that $PC = PE$ and $PD = PF$. Prove that the circumcircle of the triangle determined by the lines $AB$, $CD$ and $EF$ is tangent to the circumcircle of the triangle $ABP$.

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.

Zadan je konveksan četverokut $ABCD$ koji nije paralelogram. Neka pravac koji prolazi kroz polovišta dijagonala četverokuta siječe stranice $\overline{AB}$ i $\overline{CD}$ redom u točkama $M$ i $N$. Dokaži da trokuti $ABN$ i $CDM$ imaju jednake površine.

Dan je tetivni četverokut $ABCD$. Simetrala dužine $\overline{BC}$ siječe dužinu $\overline{AB}$ u točki $E$. Kružnica koja prolazi točkom $E$, vrhom $C$ i polovištem $F$ stranice $\overline{BC}$ siječe dužinu $\overline{CD}$ u točki $G$. Dokaži da su pravci $AD$ i $FG$ međusobno okomiti.

Neka je $A_1A_2A_3A_4$ konveksan četverokut, $S$ sjecište njegovih dijagonala. Označimo sa $s_k$ površinu trokuta $A_kSA_{k+1}$, ($A_5 = A_1$), $k = 1, 2, 3, 4$. Dokažite da je $$s_2^2 = s_1 s_3 \quad \text{i} \quad 2 s_4 = s_1 + s_3$$ ako i samo ako je $A_1A_2A_3A_4$ paralelogram.

Dan je četverokut $ABCD$ s kutovima $\alpha = 60^\circ$, $\beta = 90^\circ$, $\gamma = 120^\circ$. Dijagonale $\overline{AC}$ i $\overline{BD}$ sijeku se u točki $S$, pri čemu je $2|BS| = |SD| = 2d$. Iz polovišta $P$ dijagonale $\overline{AC}$ spuštena je okomica $\overline{PM}$ na dijagonalu $\overline{BD}$, a iz točke $S$ okomica $\overline{SN}$ na $\overline{PB}$.

Dokaži:

(a) $|MS| = |NS| = \dfrac{d}{2}$;

(b) $|AD| = |DC|$;

(c) $P(ABCD) = \dfrac{9d^2}{2}$.

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.

Neka je $ABCD$ konveksan četverokut takav da je $|AB| = 4$, $|BC| = 7$, $|AD| = 5$, $|\measuredangle CBA| = 90^\circ$, te su kutovi $\measuredangle ADC$ i $\measuredangle DCB$ šiljasti i međusobno sukladni. Odredi duljinu dužine $\overline{CD}$.

Polukrug promjera $\overline{PQ}$ upisan je u pravokutnik $ABCD$ i dira njegove stranice $\overline{AB}$ i $\overline{AD}$. Pritom se točka $P$ nalazi na stranici $\overline{BC}$, a točka $Q$ na stranici $\overline{CD}$. Ako je $|BP| = 2$ i $|DQ| = 1$, odredi $|PQ|$.

Duljine stranica četverokuta su cjelobrojne, a svaka od njih je djelitelj zbroja preostalih triju duljina. Dokaži da su bar dvije stranice tog četverokuta sukladne.

U konveksnom četverokutu $ABCD$ vrijedi $|AD| = |CD|$ i $\measuredangle ADC = 90°$. Ako je $|AB| = a$, $|BC| = b$, $|BD| = d$, $\measuredangle ABC = \beta$, dokaži da vrijedi $$2d^2 = a^2 + b^2 + 2ab \sin \beta.$$

U konveksnom četverokutu $ABCD$ vrijedi $\measuredangle BAD = 50°$, $\measuredangle ADB = 80°$ i $\measuredangle ACB = 40°$.

Ako je $\measuredangle DBC = 30° + \measuredangle BDC$, izračunaj $\measuredangle BDC$.

U četverokutu $ABCD$ je $\measuredangle DBC = \measuredangle DCB = 50°$ i $\measuredangle DAB = \measuredangle ABC = \measuredangle BDC$. Dokaži da je $AC \perp BD$.

U četverokutu $ABCD$ vrijedi $|\measuredangle ABC| = 90°$, $|\measuredangle BCD| = 120°$ i $|\measuredangle CDA| = 90°$. Neka je $M$ sjecište dijagonala $\overline{AC}$ i $\overline{BD}$. Ako je $|BM| = 1$ i $|MD| = 2$, odredi površinu četverokuta $ABCD$.

Neka je $ABCD$ konveksni četverokut i neka su $P$ i $Q$ redom točke na njegovim stranicama $\overline{BC}$ i $\overline{CD}$ takve da je $\measuredangle BAP = \measuredangle DAQ$. Dokažite da trokuti $ABP$ i $ADQ$ imaju jednake površine ako i samo ako je spojnica njihovih ortocentara okomita na pravac $AC$.

Konveksni četverokut podijeljen je dijagonalama na četiri trokuta čije su upisane kružnice sukladne. Dokaži da je taj četverokut romb.

Neka je $ABCD$ konveksni četverokut takav da vrijedi

$$\measuredangle BAD = 90^\circ, \quad \measuredangle BAC = 2\measuredangle BDC \quad \text{i} \quad \measuredangle DBA + \measuredangle DCB = 180^\circ.$$

Odredi mjeru kuta $\measuredangle DBA$.

U konveksnom četverokutu $ABCD$ vrijedi $$\measuredangle BAC = 48°, \quad \measuredangle CAD = 66°, \quad \measuredangle CBD = \measuredangle DBA.$$ Odredi kut $\measuredangle BDC$.