Let $ABC$ be a triangle with orthocenter $H$. Let the circle $\omega$ have $BC$ as the diameter. Draw tangents $AP$, $AQ$ to the circle $\omega $ at the point $P, Q$ respectively. Prove that $ P,H,Q$ lie on the same line .

The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.

Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.

In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.

Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear. by I. Dmitriev

Let $ABDE, BCFZ$ and $CAKL$ be three arbitrary rectangles constructed outside a triangle $ABC$. Let $EF$ meet $ZK$ at $M$, and $N$ be the intersection of the lines through $F,Z$ perpendicular to $FL,ZD$. Prove that $A,M,N$ are collinear. Kostas Vittas

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.

Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$. [i](Proposed by Walther Janous)[/i]

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

Let $ABCD$ be a cyclic convex quadrilateral. Let $E$ be the intersection of lines $AB,CD$. $P$ is the intersection of line passing $B$ and perpendicular to $AC$, and line passing $C$ and perpendicular to $BD$. $Q$ is the intersection of line passing $D$ and perpendicular to $AC$, and line passing $A$ and perpendicular to $BD$. Prove that three points $E, P,Q$ are collinear.

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \ne D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$. Show that $A, X$, and $Y$ are collinear.

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$. Tangent of $\Gamma$ at $D$ intersects lines $BC$, $CA$, $AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects the circle $\omega$ again at $F$. Prove that points $D,E,F$ are collinear. (Saudi Arabia)

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

Segments connect vertices $A, B, C$ of $\vartriangle ABC$ with respective points $A_1, B_1, C_1$ on the opposite sides of the triangle. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ do not belong to one straight line.

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

Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear. (Malik Talbi)

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $\triangle ABC$ be a triangle and $H$ its orthocenter. We draw the circumference $\mathcal{C}_1$ that passes through $H$ and its tangent to $BC$ at $B$ and the circumference $\mathcal{C}_2$ that passes through $H$ and its tangent to $BC$ at $C$. If $\mathcal{C}_1$ cuts line $AB$ again at $X$ and $\mathcal{C}_2$ cuts line $AC$ again at $Y$. Prove that $X,Y$ and $H$ are collinear.

A diameter $AK$ is drawn for the circumscribed circle $\omega$ of an acute-angled triangle $ABC$, an arbitrary point $M$ is chosen on the segment $BC$, the straight line $AM$ intersects $\omega$ at point $Q$. The foot of the perpendicular drawn from $M$ on $AK$ is $D$, the tangent drawn to the circle $\omega$ through the point $Q$, intersects the straight line $MD$ at $P$. A point $L$ (different from $Q$) is chosen on $\omega$ such that $PL$ is tangent to $\omega$. Prove that points $L$, $M$ and $K$ lie on the same line.

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly. 1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$. 2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.