In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.

Let $ABC$ be a triangle and $P$ be the midpoint of arc $BAC$ of circumcircle of triangle $ABC$ with orthocenter $H$. Let $Q, S$ be points such that $HAPQ$ and $SACQ$ are parallelograms. Let $T$ be the midpoint of $AQ$, and $R$ be the intersection point of the lines $SQ$ and $PB$. Prove that $AB$, $SH$ and $TR$ are concurrent. [i]Proposed by Dominik Burek - Poland[/i]

Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $. (Kivva Bogdan)

Let $ABC$ be an acute scalene triangle such that $AB <AC$. The midpoints of sides $AB$ and $AC$ are $M$ and $N$, respectively. Let $P$ and $Q$ be points on the line $MN$ such that $\angle CBP = \angle ACB$ and $\angle QCB = \angle CBA$. The circumscribed circle of triangle $ABP$ intersects line $AC$ at $D$ ($D\ne A$) and the circumscribed circle of triangle $AQC$ intersects line $AB$ at $E$ ($E \ne A$). Show that lines $BC, DP,$ and $EQ$ are concurrent. Nicolás De la Hoz, Colombia

Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency. Nguyễn Văn Linh

Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency. ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$. Nguyễn Minh Hà

An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.

Let $[ABC]$ be a triangle and $X, Y$ and $Z$ points on the sides $[AB], [BC]$ and $[AC]$, respectively. Prove that circumcircles of triangles $AXZ, BXY$ and $CYZ$ intersect at a point.

a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$. b) for $n=3$ the same problem.

A quadrilateral $ABCD$ is such that the sides $AB$ and $DC$ are parallel, and $|BC| =|AB| + |CD|$. Prove that the angle bisectors of the angles $\angle ABC$ and $\angle BCD$ intersect at right angles on the side $AD$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let three circles be externally tangent in pairs, with parallel diameters $A_1A_2, B_1B_2, C_1C_2$ (i.e. each of the quadrilaterals $A_1B_1B_2A_2$ and $A_1C_1C_2A_2$ is a parallelogram or trapezoid, which segment $A_1A_2$ is the base). Prove that $A_1B_2, B_1C_2, C_1A_2$ intersect at one point. (Yuri Biletsky )

Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria: a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$, b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$, c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$. Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point. [i]Proposed by Farras Mohammad Hibban Faddila[/i]

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.

Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ intersects $\Gamma$ again at $F$. Suppose $B$, $D$, $F$, $E$ are concyclic. Prove that $AC$, $BF$, $DE$ are concurrent.

Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$. Prove that $ AA',BB'$ and $ CC'$ are concurrent. [img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]

The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.

A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$. (b) Prove that $DL,EM$, and $FN$ are concurrent.

The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.

Let $ABC$ be a triangle, and $A_1$, $B_1$, $C_1$ points on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $A_1B_1C_1$ is equilateral. Let $I_1$ and $\omega_1$ be the incenter and the incircle of $AB_1C_1$. Define $I_2$, $\omega_2$ and $I_3$, $\omega_3$ similarly, with respect to the triangles $BA_1C_1$ and $CA_1B_1$, respectively. Let $l_1 \neq BC$ be the external tangent line to $\omega_2$ and $\omega_3$. Define $l_2$ and $l_3$ similarly, with respect to the pairs $\omega_1$, $\omega_3$ and $\omega_1$, $\omega_2$. Knowing that $A_1I_2 = A_1I_3$, show that the lines $l_1$, $l_2$, $l_3$ are concurrent.

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

Let $ABCD \dots KLMN$ be a regular polygon with $14$ sides. Show that the diagonals $AE$, $BG$, and $CK$ are concurrent.

In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.