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Na polukružnici s promjerom $\overline{AB}$ dane su točke $K$ i $L$. Simetrala dužine $\overline{AB}$ siječe dužinu $\overline{KL}$ u točki $U$ i pritom su točke $A$ i $K$ s jedne strane te simetrale, a $B$ i $L$ s druge. Neka je $N$ nožište okomice iz sjecišta pravaca $AK$ i $BL$ na pravac $AB$, a $V$ točka na pravcu $KL$ takva da je $\measuredangle VAU = \measuredangle VBU$.

Dokaži da su pravci $NV$ i $KL$ međusobno okomiti.

Dan je jednakokračni trokut $ABC$ s osnovicom $\overline{AB}$. Točka $P$ na stranici $\overline{AC}$ i točka $Q$ na stranici $\overline{BC}$ odabrane su tako da je $|AP| + |BQ| = |PQ|$. Paralela s pravcem $BC$ kroz polovište dužine $\overline{PQ}$ siječe dužinu $\overline{AB}$ u točki $N$. Kružnica opisana trokutu $PNQ$ siječe pravac $AC$ u točkama $P$ i $K$, a pravac $BC$ u točkama $Q$ i $L$. Ako je točka $R$ sjecište pravaca $PL$ i $QK$, dokaži da je pravac $PQ$ okomit na pravac $CR$.

Dan je trokut $ABC$ takav da je $|AB| < |AC|$. Na stranicama $\overline{AB}$ i $\overline{BC}$, redom su dane točke $P$ i $Q$ takve da su pravci $AQ$ i $CP$ okomiti, a kružnica upisana trokutu $ABC$ dira dužinu $\overline{PQ}$. Pravac $CP$ siječe kružnicu opisanu trokutu $ABC$ u točkama $C$ i $T$.

Ako se pravci $CA$, $PQ$ i $BT$ sijeku u jednoj točki, dokaži da je kut $\measuredangle CAB$ pravi.

Dana je kružnica promjera $\overline{AB}$. Na toj kružnici, s različitih strana pravca $AB$, nalaze se točke $C$ i $D$ takve da vrijedi $|AC| < |BC|$ i $|AC| < |AD|$. Točka $P$ pripada dužini $\overline{BC}$ te vrijedi $\measuredangle CAP = \measuredangle ABC$. Okomica iz točke $C$ na pravac $AB$ siječe pravac $BD$ u točki $Q$. Pravci $PQ$ i $AD$ sijeku se u točki $R$, a pravci $PQ$ i $CD$ u točki $T$.

Ako je $|AR| = |RQ|$, dokaži da su pravci $AT$ i $PQ$ međusobno okomiti.

Dan je trokut $ABC$ takav da je $|AC| = |BC|$ i točka $D$ na stranici $\overline{AB}$ takva da je $|AD| < |BD|$. Točke $P$ i $Q$ su redom nožišta okomica iz točke $D$ na stranice $\overline{AC}$ i $\overline{BC}$. Simetrala dužine $\overline{PQ}$ siječe $\overline{CP}$ u točki $E$. Kružnice opisane trokutima $ABC$ i $PQC$ sijeku se u točkama $C$ i $F$.

Ako su točke $E$, $F$ i $Q$ kolinearne, dokaži da je $\measuredangle ACB$ pravi kut.

U trokutu $ABC$ vrijedi $|AB| \neq |AC|$ i upisana kružnica dira stranice $\overline{BC}$, $\overline{AC}$ i $\overline{AB}$ redom u točkama $D$, $E$ i $F$. Okomica iz točke $D$ na pravac $EF$ sijeće stranicu $\overline{AB}$ u točki $G$, a kružnice opisane trokutima $AEF$ i $ABC$ se sijeku u točkama $A$ i $T$.

Dokaži da su pravci $TG$ i $TF$ okomiti.

Neka je $ABC$ raznostranični šiljastokutni trokut u kojem je $|AB| > |BC|$. Kružnica promjera $\overline{AC}$ sijeće stranicu $\overline{AB}$ u točki $X$. Na toj kružnici nalazi se točka $Y$ takva da je $CA$ simetrala kuta $\measuredangle YCB$. Neka je $D$ nožište okomice iz $B$ na $AY$. Dužine $\overline{AC}$ i $\overline{XY}$ sijeku se u točki $E$, a dužine $\overline{AC}$ i $\overline{BD}$ u točki $K$. Ako je $T$ točka na stranici $\overline{AB}$ takva da je $TK$ simetrala kuta $\measuredangle ETD$, dokaži da je $TK$ okomito na $AB$.

Neka je $M$ polovište stranice $\overline{AB}$ trokuta $ABC$ u kojem je $|BC| > |AC|$, te neka je $N$ nožište okomice iz točke $A$ na dužinu $\overline{CM}$. Neka je $P$ točka na pravcu $AN$ takva da je $PB$ okomito na $CB$.

Ako vrijedi $\measuredangle CPB = \measuredangle CBA$, dokaži da je $\measuredangle BAC = 90°$.

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$, (b) the interior of $\triangle OAB$?

$P$ is a given point inside a given sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U, V$, and $W$; $Q$ denotes the vertex diagonally opposite to $P$ in the parallelepiped determined by $PU$, $PV$, and $PW$. Find the locus of $Q$ for all such triads of rays from $P$.

$P$ is a point inside a given triangle $ABC$. $D, E, F$ are the feet of the perpendiculars from $P$ to the lines $BC, CA, AB$ respectively. Find all $P$ for which $$\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$$ is least.

In an acute-angled triangle $ABC$ the interior bisector of the angle $A$ intersects $BC$ at $L$ and intersects the circumcircle of $ABC$ again at $N$. From point $L$ perpendiculars are drawn to $AB$ and $AC$, the feet of these perpendiculars being $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.

$ABC$ is an isosceles triangle with $AB = AC$. Suppose that

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear.

Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.

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.

Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition:

for any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an axis of symmetry for $S$.

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC>60^{\circ}$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $O$ parallel to $AD$ meets $AC$ at $J$. Show that $J$ is the incenter of triangle $CEF$.

$ABCD$ 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 triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.

Let $ABC$ be a triangle with $\angle BCA = 90°$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK = ML$.

Let $\Gamma$ be the circumcircle of acute-angled triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively, such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect the minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$, respectively. Prove that the lines $DE$ and $FG$ are parallel (or are the same line).

Let $I$ be the incentre of acute triangle $ABC$ with $AB \neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

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 $ABC$ be an isosceles triangle with $AC = BC$. Let $N$ be a point inside the triangle such that $2\angle ANB = 180° + \angle ACB$. Let $D$ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$.

Show that the lines $DP$ and $AN$ are perpendicular.

Let $ABC$ be an acute-angled triangle with $\measuredangle BAC > 45°$ and with circumcentre $O$. The point $P$ lies in its interior such that the points $A, P, O, B$ lie on a circle and $BP$ is perpendicular to $CP$. The point $Q$ lies on the segment $BP$ such that $AQ$ is parallel to $PO$.

Prove that $\measuredangle QCB = \measuredangle PCO$.

Let $ABC$ be a triangle with $AB \neq AC$. The points $K, L, M$ are the midpoints of the sides $BC, CA, AB$, respectively. The inscribed circle of $ABC$ with centre $I$ touches the side $BC$ at point $D$. The line $g$, which passes through the midpoint of segment $ID$ and is perpendicular to $IK$, intersects the line $LM$ at point $P$.

Prove that $\measuredangle PIA = 90^{\circ}$.

Let $ABC$ be a triangle. The internal bisector of $\angle ABC$ intersects the side $AC$ at $L$ and the circumcircle of triangle $ABC$ again at $W \neq B$. Let $K$ be the perpendicular projection of $L$ onto $AW$. The circumcircle of triangle $BLC$ intersects line $CK$ again at $P \neq C$. Lines $BP$ and $AW$ meet at point $T$. Prove that $AW = WT$.

Let $ABC$ be an acute-angled triangle such that $AB < AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD = \angle MPB$.

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.

Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.

Let $ABC$ be a triangle with incenter $I$. The incircle $\omega$ of $ABC$ is tangent to the line $BC$ at point $D$. Denote by $E$ and $F$ the points satisfying $AI \parallel BE \parallel CF$ and $\angle BEI = \angle CFI = 90°$. Lines $DE$ and $DF$ intersect $\omega$ again at points $E'$ and $F'$, respectively. Prove that $E'F' \perp AI$.

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.

Let $ABC$ be an acute triangle with $AB < AC$. Let $J$ be the center of the $A$-excircle of $ABC$. Let $D$ be the projection of $J$ on line $BC$. The internal bisectors of angles $BDJ$ and $JDC$ intersect lines $BJ$ and $JC$ at $X$ and $Y$, respectively. Segments $XY$ and $JD$ intersect at $P$. Let $Q$ be the projection of $A$ on line $BC$. Prove that the internal angle bisector of $\measuredangle QAP$ is perpendicular to line $XY$.

Remark. The $A$-excircle of the triangle $ABC$ is the circle outside the triangle which is tangent to the lines $AB$, $AC$, and the line segment $BC$.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC, ABM, ACM$, respectively. Let $P, Q$ be points on the lines $MK, MJ$, respectively, such that $\angle AJP = \angle ABC$ and $\angle AKQ = \angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

Dokažite da su težišnice iz vrhova $A$ i $B$ trokuta $ABC$ međusobno okomite ako i samo ako za duljine stranica vrijedi jednakost $$|BC|^2 + |AC|^2 = 5|AB|^2.$$

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.

Dužina $\overline{AB}$ je promjer kružnice sa središtem $O$. Na kružnici je dana točka $C$ takva da je $OC$ okomito na $AB$. Na kraćem luku $\widehat{BC}$ odabrana je točka $P$. Pravci $CP$ i $AB$ sijeku se u točki $Q$, a točka $R$ je sjecište pravca $AP$ i okomice kroz $Q$ na pravac $AB$.

Dokaži da je $|BQ| = |QR|$.

U jednakokračnom trokutu $ABC$ vrijedi $|AB| = |AC|$ i $\measuredangle BAC < 60°$. Neka je točka $D$ na dužini $\overline{AC}$ takva da je $\measuredangle DBC = \measuredangle BAC$, neka je $E$ sjecište simetrale dužine $\overline{BD}$ i paralele s $BC$ kroz točku $A$ te neka je $F$ točka na pravcu $AC$ takva da se $A$ nalazi između $C$ i $F$ i vrijedi $|AF| = 2|AC|$.

(a) Dokaži da su pravci $BE$ i $AC$ paralelni.

(b) Dokaži da se okomica iz $F$ na $AB$ i okomica iz $E$ na $AC$ sijeku na pravcu $BD$.

U (b) dijelu zadatka dozvoljeno je korištenje tvrdnje iz (a) čak i ako nije dokazana.

Dan je trokut $ABC$ u kojem je $\measuredangle BAC = 45°$, $|AB| = 4$, $|AC| = 3\sqrt{2}$. Neka su $\overline{AD}$ i $\overline{BE}$ visine tog trokuta. Okomica na $\overline{AB}$ kroz točku $E$ siječe dužinu $\overline{AD}$ u točki $P$.

Odredi $|EP|$.

Neka je $ABC$ pravokutni trokut s pravim kutom u vrhu $C$. Neka je $P$ točka takva da je kut $\measuredangle ABP$ je pravi, da vrijedi $|BP| = |BC|$ te da su točke $P$ i $C$ su na suprotnim stranama pravca $AB$. Dokaži da je pravac $CP$ okomit na simetralu kuta $\measuredangle BAC$.

Jedan kut pravokutnog trokuta $\Delta ABC$ iznosi $30^\circ$, a kraća kateta duljine je $3$ cm. U polovištu $S$ hipotenuze $\overline{AB}$ podignuta je okomica na hipotenuzu i njezino sjecište s duljom katetom označeno je s $D$. Odredite duljinu dužine $\overline{SD}$.

Osnovica $\overline{BC}$ je najdulja stranica jednakokračnog trokuta $ABC$. Neka je $M$ točka na stranici $\overline{BC}$ takva da je $|BM| = |AB|$. Nožište okomice iz točke $M$ na $\overline{AB}$ je točka $N$. Dokaži da trokut $BMN$ i četverokut $ACMN$ imaju jednake površine i jednake opsege.

Neka je $M$ polovište stranice $\overline{BC}$ paralelograma $ABCD$. Ako je $E$ nožište okomice iz točke $D$ na pravac $AM$, dokaži da vrijedi $|CD| = |CE|$.

Neka je $D$ nožište visine iz vrha $A$ u šiljastokutnome trokutu $ABC$. Točke $E$ i $F$ su redom nožišta okomica iz točke $D$ na $AB$ i $AC$, a točke $G$ i $H$ redom su nožišta okomica iz $E$ i $F$ na $AD$. Ako je $|AH| = |HG| = |GD| = 2$, odredi površinu trokuta $ABC$.

Na stranicama $\overline{AB}$ i $\overline{BC}$ kvadrata $ABCD$ izabrane su točke $E$ i $F$, tim redom, takve da je $|BE| = |BF|$. Neka je $\overline{BN}$ visina trokuta $BCE$. Dokažite da je trokut $DNF$ pravokutan.

Nad stranicama $\overline{AC}$ i $\overline{BC}$ šiljastokutnog trokuta $ABC$ s vanjske strane konstruirani su kvadrati $ACXE$ i $CBDY$. Dokažite da se pravci $AD$ i $BE$ sijeku na visini iz vrha $C$ trokuta $ABC$.

Dana je polukružnica nad promjerom $\overline{AB}$ i na njoj točke $C$ i $D$ tako da vrijedi:

a) točka $C$ pripada luku $\widehat{AD}$;

b) $\measuredangle CSD$ je pravi, pri čemu je $S$ središte dužine $\overline{AB}$.

Neka je $E$ sjecište pravaca $AC$ i $BD$, a $F$ sjecište $AD$ i $BC$. Dokažite da je $|EF| = |AB|$.

Upisana kružnica dodiruje stranice $AB$ i $AC$ trokuta $ABC$ u točkama $M$ i $N$. Neka je $P$ sjecište pravca $MN$ i simetrale kuta $\measuredangle ABC$. Dokaži da je $BP \perp CP$.

Trapez $ABCD$ s osnovicama $\overline{AB}$ i $\overline{CD}$ ima opisanu kružnicu $k$. Njegove dijagonale međusobno su okomite i sijeku se u točki $S$. Odredi omjer površine kruga omedenog kružnicom $k$ i zbroja površina trokuta $ABS$ i $CDS$.

Na kružnici $k$ nalaze se točke $A$ i $B$, a na manjem luku $\widehat{AB}$ točka $P$. Neka su $Q$ i $R$ točke na $k$, različite od $P$, takve da je $|AP| = |AQ|$ i $|BP| = |BR|$. Neka je $T$ sjecište pravaca $AR$ i $BQ$. Dokaži da su pravci $PT$ i $AB$ međusobno okomiti.