Found problems: 250
2013 Singapore Junior Math Olympiad, 2
In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.
1981 Bundeswettbewerb Mathematik, 2
Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.
2022 Korea National Olympiad, 2
In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.
2009 Thailand Mathematical Olympiad, 3
Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot MC + MA \cdot CD = MB \cdot MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.
2011 Sharygin Geometry Olympiad, 11
The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.
2004 District Olympiad, 2
Let $ABC$ be a triangle and $D$ a point on the side $BC$. The angle bisectors of $\angle ADB ,\angle ADC$ intersect $AB ,AC$ at points $M ,N$ respectively. The angle bisectors of $\angle ABD , \angle ACD$ intersects $DM , DN$ at points $K , L$ respectively. Prove that $AM = AN$ if and only if $MN$ and $KL$ are parallel.
2018 Saudi Arabia IMO TST, 3
Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.
2018 Bosnia And Herzegovina - Regional Olympiad, 4
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$
1976 Bundeswettbewerb Mathematik, 3
A circle is divided by $2n$ points into $2n$ equal arcs. Let $P_1, P_2, \ldots, P_{2n}$ be an arbitrary permutation of the $2n$ division points. Prove that the polygonal line $P_1 P_2 \cdots P_{2n} P_1$ contains at least two parallel segments.
2022 Sharygin Geometry Olympiad, 5
Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral. The line parallel to $BD$ passing through $A$ meets the line parallel to $AC$ passing through $B$ at $E$. The circumcircle of triangle $ABE$ meets the lines $EC$ and $ED$, again, at $F$ and $G$, respectively. Prove that the lines $AB, CD$ and $FG$ are either parallel or concurrent.
2013 Ukraine Team Selection Test, 6
Six different points $A, B, C, D, E, F$ are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne P', Q \ne Q', R \ne R'$, and prove that the lines $PP', QQ'$ and $RR'$ intersect at one point or are parallel.
2019 Romanian Master of Mathematics Shortlist, G2
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.
Giorgi Arabidze, Georgia,
1973 IMO Shortlist, 6
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
2013 Czech And Slovak Olympiad IIIA, 5
Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.
2020 Macedonia Additional BMO TST, 2
Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent:
$i) \angle BLA= \angle CKD$
$ii) AD \parallel BC $
2004 Regional Olympiad - Republic of Srpska, 2
Let $ABC$ be a triangle, $T$ its centroid and $S$ its incenter.
Prove that the following conditions are equivalent:
(1) line $TS$ is parallel to one side of triangle $ABC$,
(2) one of the sides of triangle $ABC$ is equal to the half-sum of the other two sides.
2016 Flanders Math Olympiad, 1
In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles.
[img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]
Kharkiv City MO Seniors - geometry, 2013.11.4
In the triangle $ABC$, the heights $AA_1$ and $BB_1$ are drawn. On the side $AB$, points $M$ and $K$ are chosen so that $B_1K\parallel BC$ and $A_1 M\parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.
Swiss NMO - geometry, 2018.6
Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.
2018 Bundeswettbewerb Mathematik, 3
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$.
Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.
2016 Abels Math Contest (Norwegian MO) Final, 3b
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.
Durer Math Competition CD Finals - geometry, 2013.D3
The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.
2001 All-Russian Olympiad Regional Round, 10.2
In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.
1994 ITAMO, 4
Let $ABC$ be a triangle contained in one of the halfplanes determined by a line $r$. Points $A',B',C'$ are the reflections of $A,B,C$ in $r,$ respectively. Consider the line through $A'$ parallel to $BC$, the line through $B'$ parallel to $AC$ and the line through $C'$ parallel to $AB$. Show that these three lines have a common point.