Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.

A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$. Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.

Let $ABC$ be a triangle inscribed in a circle ($\omega$) and $I$ is the incenter. Denote $D,E$ as the intersection of $AI,BI$ with ($\omega$). And $DE$ cuts $AC,BC$ at $F,G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of ($\omega$) at $A,B$ meet at $K$. Prove that three lines $AE,BD,KP$ are concurrent or parallel.

Two triangles $ABC$ and $PQR$ have the same circumcircles. Let $E_a, E_b, E_c$ be the centers of the Euler circles of triangles $PBC, QCA, RAB$. Assume that $d_a$ is a line through $Ea$ parallel to $AP$, $d_b, d_c$ are defined in the same manner. Prove that three lines $d_a, d_b, d_c$ are concurrent. Nguyễn Tiến Lâm, Trần Quang Hùng

Given triangle $ABC$. Point $B_1$ bisects the length of the broken line $ABC$ (composed of segments $AB$ and $BC$), point $C_1$ bisects the length of the broken line$ACB$, point $A_1$ bisects the length of of the broken line $CAB$. Through points $A_1$, $B_1$ and $C_1$ straight lines $\ell_A$ ,$\ell_B$, $\ell_C$ are drawn parallel to the bisectors angles $BAC$, $ABC$ and $ACB$ respectively. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ intersect at one point.

In an acute triangle $ABC$, the bisector $AL$, the altitude $BH$, and the perpendicular bisector of the side $AB$ intersect at one point. Find the value of the angle $BAC$.

Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.

Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .

Let $A_1,$ $B_1$, $C_1$ be the touchpoints of the circle inscribed in the acute triangle $ABC$ ($A_1$ is the touchpoint with the side $BC$, etc.). Let $A_2$, $B_2$, $C_2$ be the intersection points of the altitudes of triangles $AB_1C_1$, $A_1BC_1$ and $A_1B_1C$ respectively. Prove that the lines $A_1A_2$ and $B_1B_2$ and $C_1C_2$ intersect at one point.

Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.

Let $ABC$ be an acute triangle, not isoceles triangle and $(O), (I)$ be its circumcircle and incircle respectively. Let $A_1$ be the the intersection of the radical axis of $(O), (I)$ and the line $BC$. Let $A_2$ be the point of tangency (not on $BC$) of the tangent from $A_1$ to $(I)$. Points $B_1,B_2,C_1,C_2$ are defined in the same manner. Prove that (a) the lines $AA_2,BB_2,CC_2$ are concurrent. (b) the radical centers circles through triangles $BCA_2, CAB_2$ and $ABC_2$ are all on the line $OI$. Lê Phúc Lữ

Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$, and $Q'$ on the circle center $O'$). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$. [img]https://cdn.artofproblemsolving.com/attachments/b/d/bad305631571323a61b097f149a1bb6855cdc5.png[/img]

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.

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. [i]Bogdan Enescu, Romania[/i]

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel. (Serdyuk Nazar)

The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.

Three circles are constructed for the triangle $ABC $: the circle ${{w} _ {A}} $ passes through the vertices $B $ and $C $ and intersects the sides $AB $ and $ AC $ at points ${{A} _ {1}} $ and ${{A} _ {2}} $ respectively, the circle ${{w} _ {B}} $ passes through the vertices $A $ and $C $ and intersects the sides $BA $ and $BC $ at the points ${{B} _ {1}} $ and ${{B} _ {2}} $, ${{w} _ {C}} $ passes through the vertices $A $ and $B $ and intersects the sides $CA $ and $CB $ at the points ${{C} _ {1}} $ and ${{C} _ {2}} $. Let ${{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '$, ${{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '$ ta ${ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '$ is Prove that the perpendiculars, which are omitted from the points ${A} ', \, \, {B}', \, \, {C} '$ to the lines $BC $, $CA $ and $AB $ respectively intersect at one point. (Rudenko Alexander)

Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$ K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent. Michel Bataille , France

A convex hexagon $ABCDEF$ is such that $$AB=BC \quad CD=DE \quad EF=FA$$ and $$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$ Show that $AD$, $CF$ and $EB$ are concurrent.

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of acute angled triangle $ABC$. $A_2$ be the touching point of the incircle of triangle $AB_1C_1$ with $B_1C_1$, points $B_2$, $C_2$ be defined similarly. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ concur.