Let $ABCDEF$ be a hexagon inscribed in a circle in which $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines $AD, BE$ and $CF$ intersect at one point.

Points $P, Q, R$ were marked on the sides $BC, CA, AB$, respectively. Let $a$ be tangent at point $A$ to the circumcircle of triangle $AQR$, $b$ be tangent at point $B$ to the circumcircle of the triangle BPR, $c$ be tangent at point $C$ to the circumscribed circle triangle $CPQ$. Let $X$ be the point of intersection of the lines $b$ and $c, Y$ be the point the intersection of lines $c$ and $a, Z$ is the point of intersection of lines $a$ and $b$. Prove that the lines $AX, BY, CZ$ intersect at one point if and only if the lines $AP, BQ, CR$ intersect at one point.

Let $ABCD$ be a trapezium in which $AB$ is parallel to $CD$. The circles on $AD$ and $BC$ as diameters intersect at two distinct points $P$ and $Q$. Prove that the lines $PQ,AC,BD$ are concurrent.

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$

Given an acute triangle $ABC$ ($AC\ne AB$) and let $(C)$ be its circumcircle. The excircle $(C_1)$ corresponding to the vertex $A$, of center $I_a$, tangents to the side $BC$ at the point $D$ and to the extensions of the sides $AB,AC$ at the points $E,Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $(C_1)$, $H$ the orthocenter of the triangle $EDZ$ and $N$ the midpoint of segment $EZ$. The parallel line through the point $l_a$ to the line $HL$ meets the line $HI$ at the point $G$. Prove that the perpendicular line $(e)$ through the point $N$ to the line $BC$ and the parallel line $(\delta)$ through the point $G$ to the line $IL$ meet each other on the line $HI_a$.

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 $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic. (ltf0501).

In a triangle $ABC, A_1,B_1,C_1$ are the midpoints of segments $BC,CA,AB, A_2,B_2,C_2$ are the midpoints of segments $B_1C_1,C_1A_1,A_1B_1$, and $A_3,B_3,C_3$ are the incenters of triangles $B_1AC_1,C_1BA_1,A_1CB_1$, respectively. Show that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrent.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)

Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.

Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

In a cyclic convex hexagon $ABCDEF$, $AB$ and $DC$ intersect at $G$, $AF$ and $DE$ intersect at $H$. Let $M, N$ be the circumcenters of $BCG$ and $EFH$, respectively. Prove that the $BE$, $CF$ and $MN$ are concurrent.

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

Let $A_1A_2 \dots A_{13}$ and $B_1B_2 \dots B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9, B_{13}B_8$ and $A_8B_9$ are concurrent.

In a triangle $ABC$ a circle $k$ is inscribed, which is tangent to $BC$,$CA$,$AB$ in points $D,E,F$ respectively. Let point $P$ be inner for $k$. If the lines $DP$,$EP$,$FP$ intersect $k$ in points $D',E',F'$ respectively, then prove that $AD'$, $BE'$, and $CF'$ are concurrent.

In a convex quadrilateral $ABCD$, angles $B$ and $D$ are right angles. On on sides $AB$, $BC$, $CD$, $DA$ points $K$, $L$, $M$, $N$ are taken respectively so that $KN \perp AC$ and $LM \perp AC$. Prove that $KM$, $LN$ and $AC$ intersect at one point.

In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.

Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

Let$ AB$ and $AC$ be the tangents from a point $A$ to a circle $ \Omega$. Let $M$ be the midpoint of $BC$ and $P$ be an arbitrary point on this segment. A line $AP$ meets $ \Omega$ at points $D$ and $E$. Prove that the common external tangents to circles $MDP$ and $MPE$ meet on the midline of triangle $ABC$.

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.