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U šiljastokutnom trokutu $ABC$ vrijedi $|AB| > |BC|$, a točke $A_1$ i $C_1$ su redom nožišta visina iz vrhova $A$ i $C$. Neka je $D$ drugo sjecište kružnica opisanih trokutima $ABC$ i $A_1BC_1$ (različito od $B$). Neka je $Z$ sjecište tangenata na opisanu kružnicu trokuta $ABC$ u točkama $A$ i $C$, te neka se pravci $ZA$ i $A_1C_1$ sijeku u točki $X$, a pravci $ZC$ i $A_1C_1$ u točki $Y$.

Dokaži da točka $D$ leži na kružnici opisanoj trokutu $XYZ$.

Neka je $I$ središte upisane kružnice trokuta $ABC$, a točka $D$ na stranici $\overline{AC}$ takva da je $|AB| = |DB|$. Upisana kružnica trokuta $BCD$ dodiruje pravce $AC$ i $BD$ redom u točkama $E$ i $F$. Dokaži da pravac $EF$ raspolavlja dužinu $\overline{DI}$.

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.

Dana je kružnica $k$ sa središtem $O$. Neka je $\overline{AB}$ tetiva te kružnice i $M$ njeno polovište. Tangente na kružnicu $k$ u točkama $A$ i $B$ sijeku se u $T$. Pravac $\ell$ prolazi točkom $T$, siječe kraći luk $\widehat{AB}$ u točki $C$, a dulji luk $\widehat{AB}$ u točki $D$ i pritom je $|BC| = |BM|$.

Dokaži da je središte kružnice opisane trokutu $ADM$ osnosimetrično točki $O$ u odnosu na pravac $AD$.

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 $ABC$ trokut. Kružnica $k$ prolazi točkom $A$, siječe stranice $\overline{AB}$ i $\overline{AC}$ redom u točkama $D$ i $E$ (različitim od $A$), a stranicu $\overline{BC}$ u točkama $F$ i $G$ i pritom je $F$ između $B$ i $G$. Tangenta opisane kružnice trokuta $BDF$ u točki $F$ i tangenta opisane kružnice trokuta $CEG$ u točki $G$ sijeku se u točki $T$, različitoj od $A$.

Dokaži da su pravci $AT$ i $BC$ međusobno paralelni.

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.

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.

Kružnice $k_1$ i $k_2$, redom sa središitima $O_1$ i $O_2$, sijeku se u točkama $A$ i $B$. Pravac $p$ prolazi točkom $B$ i sijeće kružnicu $k_1$ još u točki $C$, a kružnicu $k_2$ još u točki $D$, pri čemu se točka $B$ nalazi između $C$ i $D$. Tangenta na kružnicu $k_1$ u točki $C$ i tangenta na kružnicu $k_2$ u točki $D$ sijeku se u točki $E$. Pravac $AE$ sijeće opisanu kružnicu trokuta $AO_1O_2$ u točkama $A$ i $F$.

Dokaži da duljina $|EF|$ ne ovisi o odabiru pravca $p$.

Neka je $O$ središte opisane kružnice $k$ trokuta $ABC$ u kojem je $|AB| > |BC|$.

Kružnica $k_1$ prolazi točkama $O$ i $B$, a pravac $AB$ joj je tangenta. Neka se kružnice $k$ i $k_1$ sijeku još i u točki $P$, $P \neq B$. Kružnica $k_2$ prolazi točkama $P$ i $C$, a pravac $AC$ joj je tangenta. Neka se kružnice $k_1$ i $k_2$ sijeku još u točki $M$, $M \neq P$.

Dokaži da je $|MP| = |MC|$.

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BC,CA,AB$, or to their extensions.

(a) Prove that the tetrahedron $SABC$ is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres exist.

A circle is inscribed in triangle $ABC$ with sides $a, b, c$. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a, b, c$).

A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles, $\gamma_1, \gamma_2, \gamma_3$, all tangent to the line $AB$. Of these, $\gamma_1$ is inscribed in $\triangle ABC$, while $\gamma_2$ and $\gamma_3$ are both tangent to $CD$ and to $\gamma$, one on each side of $CD$. Prove that $\gamma_1, \gamma_2$ and $\gamma_3$ have a second tangent in common.

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$, and $M_2$ be the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2 = \angle M_1AM_2$.

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.

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

Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If

$$\frac{AM}{AB} = t,$$

find

$$\frac{EG}{EF}$$

in terms of $t$.

In the plane let $C$ be a circle, $L$ a line tangent to the circle $C$, and $M$ a point on $L$. Find the locus of all points $P$ with the following property: there exists two points $Q, R$ on $L$ such that $M$ is the midpoint of $QR$ and $C$ is the inscribed circle of triangle $PQR$.

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB = \angle ACB + \pi/2$ and $AC \cdot BD = AD \cdot BC$.

(a) Calculate the ratio $(AB \cdot CD)/(AC \cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular.

Two circles $G_1$ and $G_2$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_1$ passes through the center of $G_2$. The line passing through the two points of intersection of $G_1$ and $G_2$ meets $G$ at $A$ and $B$. The lines $MA$ and $MB$ meet $G_1$ at $C$ and $D$, respectively.

Prove that $CD$ is tangent to $G_2$.

$AB$ is tangent to the circles $CAMN$ and $NMBD$. $M$ lies between $C$ and $D$ on the line $CD$, and $CD$ is parallel to $AB$. The chords $NA$ and $CM$ meet at $P$; the chords $NB$ and $MD$ meet at $Q$. The rays $CA$ and $DB$ meet at $E$. Prove that $PE = QE$.

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 $\omega$.

Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. Let $K$, $L$ and $M$ be the midpoints of the segments $BP$, $CQ$ and $PQ$, respectively, and let $\Gamma$ be the circle passing through $K$, $L$ and $M$. Suppose that the line $PQ$ is tangent to the circle $\Gamma$. Prove that $OP = OQ$.

Let $P$ be a point inside the triangle $ABC$. The lines $AP$, $BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ again at the points $K$, $L$ and $M$ respectively. The tangent to $\Gamma$ at $C$ intersects the line $AB$ at $S$. Suppose that $SC = SP$. Prove that $MK = ML$.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a$, $\ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a$, $\ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G$. Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC$.

Prove that $M$ is the midpoint of $ST$.

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

The excircle of triangle $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$. The excircles opposite $B$ and $C$ are similarly defined.

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 $TSH$.

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets $\Omega$ again at $K$. Prove that the line $KT$ is tangent to $\Gamma$.

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 $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\measuredangle BAC$.

Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C$, $M$, $N$ and $D$ lie on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E \neq A$. Line $AP$ intersects $\Gamma$ again at $F \neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

(The orthocentre of a triangle is the point of intersection of its altitudes.)

In a plane the circles $\mathcal{K}_1$ and $\mathcal{K}_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal{K}_1$ in $A$ intersects $\mathcal{K}_2$ again in $C$ and the tangent to $\mathcal{K}_2$ in $A$ intersects $\mathcal{K}_1$ again in $D$. Let $\mathcal{K}_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal{K}_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal{K}_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

Let $ABC$ be an acute triangle. Denote by $B_{0}$ and $C_{0}$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_{0}$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_{0}$. Show that the line $AX$ is perpendicular to $BC$.

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 $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90^{\circ}$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$. Prove that the line $XY$ is tangent to $\Omega$.

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 $ABP$.

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 $E'F' \perp AI$.

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects segments $AB$ and $AC$ again in points $D \neq A$ and $E \neq A$, respectively. Let $F$ be the intersection of $BE$ and $CD$. Let $G$ be the point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$. Similarly, let $H$ be the point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T \neq A$, independent of the choice of $\omega$, such that the circumcircle of $AGH$ passes through $T$.

Let $ABC$ be an acute triangle with $AB < AC$. Denote by $D$ the foot of the perpendicular from $A$ to $BC$. Let $E$ be the point such that $ABEC$ is a parallelogram. Let $M$ be a point inside triangle $ABC$ such that $MB = MC$. Let $F$ be the reflection of point $D$ across the tangent to the circumcircle of triangle $ADM$ at point $M$. Prove that $AF = DE$.

Zadane su kružnica i tetiva koja dijeli njezinu nutrinu na dva kružna odsječka. U njih su upisane kružnice $k_{1}$ i $k_{2}$ koje iznutra diraju kružnicu $k$, i danu tetivu diraju u istoj točki s raznih njezinih strana. Dokažite da je omjer polumjera kružnica $k_{1}$ i $k_{2}$ konstantan, tj. da ne ovisi o položaju zajedničkog dirališta s tetivom.

Kružnice $k_1$ i $k_2$ polumjera $r_1 = 6$ i $r_2 = 3$ dodiruju se izvana. Obje kružnice dodiruju iznutra kružnicu $k$ polumjera $r = 9$. Zajednička vanjska tangenta kružnica $k_1$ i $k_2$ siječe kružnicu $k$ u točkama $P$ i $Q$. Izračunajte duljinu tetive $\overline{PQ}$.

Neka je $OAB$ četvrtina kruga sa središtem $O$ polumjera $1$. Nad dužinama $\overline{OA}$ i $\overline{OB}$, kao promjerima, konstruirane su polukružnice s unutarnje strane dane četvrtine kruga. Izračunaj polumjer kružnice koja dodiruje te dvije polukružnice i luk $\widehat{AB}$.

Neka je $\overline{AC}$ promjer kružnice $k_1$ kojoj je središte u točki $B$. Kružnica $k_2$ dira pravac $AC$ u točki $B$ i kružnicu $k_1$ u točki $D$. Tangenta iz $A$ (različita od $AC$) na kružnicu $k_2$ dira tu kružnicu u točki $E$ i siječe pravac $BD$ u točki $F$. Odredi omjer $|AF| : |AB|$.

Trokutu $ABC$ upisana je kružnica koja dira stranice $\overline{AB}$, $\overline{BC}$ i $\overline{AC}$ redom u točkama $D$, $E$ i $F$. Pravac koji prolazi točkom $C$ i paralelan je s $DE$ siječe pravac $DF$ u točki $M$, a pravac koji prolazi točkom $C$ i paralelan je s $DF$ siječe pravac $DE$ u točki $N$. Dokaži da pravac $MN$ sadrži srednjicu trokuta $ABC$.

Neka je $\overline{OA}$ polumjer i $\overline{OB}$ tetiva kružnice $k$ polumjera $R$, $C$ sjecište pravca $OB$ i tangente na $k$ u točki $A$, $T$ točka na dužini $\overline{OB}$ takva da je $|OT| = |BC|$ i $T'$ projekcija od $T$ na $\overline{OA}$. Izrazite $y = |T'T|$ kao funkciju od $x = |OT'|$.

Kružnica sa središtem $O$ dira stranicu $\overline{BC}$ i produžetke stranica $\overline{AB}$ i $\overline{AC}$ trokuta $ABC$ redom u točkama $K$, $P$ i $Q$. Dužine $\overline{OB}$ i $\overline{OC}$ sijeku spojnicu $\overline{PQ}$ redom u točkama $M$ i $N$. Dokažite da je $$\frac{|QN|}{|AB|} = \frac{|MN|}{|BC|} = \frac{|MP|}{|CA|}.$$

Kružnice $\mathcal{C}_1$ i $\mathcal{C}_2$ sijeku se u točkama $A$ i $B$. Tangenta kružnice $\mathcal{C}_2$ povučena iz točke $A$ siječe kružnicu $\mathcal{C}_1$ u točki $C$, a tangenta kružnice $\mathcal{C}_1$ povučena iz točke $A$ siječe kružnicu $\mathcal{C}_2$ u točki $D$. Polupravac kroz točku $A$, koji leži unutar kuta $\measuredangle CAD$, siječe kružnicu $\mathcal{C}_1$ u točki $M$, kružnicu $\mathcal{C}_2$ u točki $N$ i kružnicu opisanu trokutu $ACD$ u točki $P$. Dokaži da je udaljenost točaka $A$ i $M$ jednaka udaljenosti točaka $N$ i $P$.

Jednakokračnom trokutu $ABC$ ($|AB| = |AC|$) opisana je kružnica. Tangente te kružnice s diralištima u točkama $A$ i $C$ sijeku se u točki $D$. Ako je $\measuredangle DBC = 30°$, dokaži da je trokut $ABC$ jednakostraničan.

Neka su $p$ i $q$ dva paralelna pravca. Kružnica $k$ dodiruje pravac $p$ u točki $A$ i siječe pravac $q$ u različitim točkama $B$ i $C$. Neka je $T$ točka na pravcu $p$ i neka dužine $\overline{TB}$ i $\overline{TC}$ sijeku kraći luk $\widehat{AC}$ redom u točkama $K$ i $L$, različitima od $B$ i $C$.

Dokaži da pravac $KL$ prolazi polovištem dužine $\overline{AT}$.