Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.

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 $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.

Given an acute triangle $ABC$, in which $\angle BAC = 60^o$. On the sides $AC$ and $AB$ take the points $T$ and $Q$, respectively, such that $CT = TQ = QB$. Prove that the center of the inscribed circle of triangle $ATQ$ lies on the side $BC$. (Dmitry Shvetsov)

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges. (Yu. Blinkov)

The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.

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 $\Omega$.

Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$ in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent. (D.Shvetsov, A.Zaslavsky)

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 $\vartriangle ABC$ and a point $P$ inside that triangle. The parallelograms $CPBL$, $APCM$ and $BPAN$ are constructed. Prove that $AL$, $BM$ and $CN$ pass through one point $S$, and that $S$ is the midpoint of $AL$, $BM$ and $CN$.

Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle internal tangent to $(O)$ at $A_1$ and also tangent to segments $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B_1, C_1, B_c , B_a, C_a, C_b$ similarly. 1. Prove that $AA_1, BB_1, CC_1$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear. 2. Prove that the circle $(I)$ is inscribed in the hexagon with $6$ vertices $A_b,A_c , B_c , B_a, C_a, C_b$.

What condition should the angles of triangle $ ABC $ satisfy so that the bisector of angle $ A $, the median drawn from vertex $ B $ and the altitude drawn from vertex $ C $ intersect at one point?

Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.

Let $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent. Nguyễn Văn Linh

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.

A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases of $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. Prove that the point $L$ lies on the diagonal $BD$.

$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ so that $P$ lies between $B$ and $Q$. The rays $AP$ and $AQ$ divide the angle $BAC$ into three equal parts. It is known that the triangle $APQ$ is acute-angled. Denote by $B_1,P_1,Q_1,C_1$ the projections of points $B,P,Q,C$ onto the lines $AP,AQ,AP,AQ$, respectively. Prove that lines $B_1P_1$ and $C_1Q_1$ meet on line $BC$.

Let $ABC$ be an arbitrary triangle, and let $M, N, P$ be any three points on the sides $BC, CA, AB$ such that the lines $AM, BN, CP$ concur. Let the parallel to the line $AB$ through the point $N$ meet the line $MP$ at a point $E$, and let the parallel to the line $AB$ through the point $M$ meet the line $NP$ at a point $F$. Then, the lines $CP, MN$ and $EF$ are concurrent. [hide=MOP 97 problem]Let $ABC$ be a triangle, and $M$, $N$, $P$ the points where its incircle touches the sides $BC$, $CA$, $AB$, respectively. The parallel to $AB$ through $N$ meets $MP$ at $E$, and the parallel to $AB$ through $M$ meets $NP$ at $F$. Prove that the lines $CP$, $MN$, $EF$ are concurrent. [url=https://artofproblemsolving.com/community/c6h22324p143462]also[/url][/hide]

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 $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively. Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.

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.