Found problems: 254
1979 Chisinau City MO, 181
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
2011 Greece JBMO TST, 4
Let $ABC$ be an acute and scalene triangle with $AB<AC$, inscribed in a circle $c(O,R)$ (with center $O$ and radius $R$). Circle $c_1(A,AB)$ intersects side $BC$ at point $E$ and circle $c$ at point $F$. $EF$ intersects for the second time circle $c$ at point $D$ and side $AC$ at point $M$. $AD$ intersects $BC$ at point $K$. Circumcircle of triangle $BKD$ intersects $AB$ at point $L$ . Prove that points $K,L,M$ lie on a line parallel to $BF$.
2016 Singapore Junior Math Olympiad, 3
In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.
2023 Austrian MO Beginners' Competition, 2
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]
2021 Austrian MO National Competition, 5
Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.
(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.
(b) Show that in all other cases the four points thus obtained lie on one circle.
(Theresia Eisenkölbl)
2015 Indonesia MO Shortlist, G2
Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.
Ukraine Correspondence MO - geometry, 2020.11
The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.
2014 Oral Moscow Geometry Olympiad, 4
The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$ at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point $C'$. Points $A'$ and $B'$ are defined similarly. Prove that $A', B'$ and $C'$ lie on one line perpendicular to the line $MH$.
Ukraine Correspondence MO - geometry, 2016.7
The circle $\omega$ inscribed in an isosceles triangle $ABC$ ($AC = BC$) touches the side $BC$ at point $D$ .On the extensions of the segment $AB$ beyond points $A$ and $B$, respectively mark the points $K$ and $L$ so that $AK = BL$, The lines $KD$ and $LD$ intersect the circle $\omega$ for second time at points $G$ and $H$, respectively. Prove that point $A$ belongs to the line $GH$.
2021 Dutch IMO TST, 3
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
1959 Poland - Second Round, 6
From a point $ M $ on the surface of a sphere, three mutually perpendicular chords $ MA $, $ MB $, $ MC $ are drawn. Prove that the segment joining the point $ M $ with the center of the sphere intersects the plane of the triangle $ ABC $ at the center of gravity of this triangle.
2016 NZMOC Camp Selection Problems, 6
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $P \ne E$ be the point of tangency of the circle with radius $HE$ centred at $H$ with its tangent line going through point $C$, and let $Q \ne E$ be the point of tangency of the circle with radius $BE$ centred at $B$ with its tangent line going through $C$. Prove that the points $D, P$ and $Q$ are collinear.
2018 Chile National Olympiad, 6
Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.
2012 Belarus Team Selection Test, 2
Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear.
(A. Voidelevich)
2018-IMOC, G2
Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.
Geometry Mathley 2011-12, 3.3
A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$.
a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear.
b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent.
Trần Quang Hùng
2005 Bosnia and Herzegovina Team Selection Test, 4
On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.
Ukraine Correspondence MO - geometry, 2013.7
An arbitrary point $D$ is marked on the hypotenuse $AB$ of a right triangle $ABC$. The circle circumscribed around the triangle $ACD$ intersects the line $BC$ at the point $E$ for the second time, and the circle circumscribed around the triangle $BCD$ intersects the line $AC$ for the second time at the point $F$. Prove that the line $EF$ passes through the point $D$.
2017 Romania Team Selection Test, P4
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.
2014 Costa Rica - Final Round, 1
Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]
2019 Novosibirsk Oral Olympiad in Geometry, 4
Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]
2022 Federal Competition For Advanced Students, P2, 5
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.
[i](Walther Janous)[/i]
2004 Junior Tuymaada Olympiad, 3
Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.
2017 Brazil Team Selection Test, 3
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.
2009 All-Russian Olympiad Regional Round, 10.4
Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.