Found problems: 333
2013 Brazil National Olympiad, 6
The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$.
1954 Moscow Mathematical Olympiad, 279
Given four straight lines, $m_1, m_2, m_3, m_4$, intersecting at $O$ and numbered clockwise with $O$ as the center of the clock, we draw a line through an arbitrary point $A_1$ on $m_1$ parallel to $m_4$ until the line meets $m_2$ at $A_2$. We draw a line through $A_2$ parallel to $m_1$ until it meets $m_3$ at $A_3$. We also draw a line through $A_3$ parallel to $m_2$ until it meets $m_4$ at $A_4$. Now, we draw a line through$ A_4$ parallel to $m_3$ until it meets $m_1$ at $B$. Prove that
a) $OB< \frac{OA_1}{2}$ .
b) $OB \le \frac{OA_1}{4}$ .
[img]https://cdn.artofproblemsolving.com/attachments/5/f/5ea08453605e02e7e1253fd7c74065a9ffbd8e.png[/img]
2006 Chile National Olympiad, 6
Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $
a) Show that $ O_1O_2 = BC $.
b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent
Oliforum Contest I 2008, 3
Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$.
Prove that $ AA',BB'$ and $ CC'$ are concurrent.
[img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]
2011 Sharygin Geometry Olympiad, 21
On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.
1961 Polish MO Finals, 5
Four lines intersecting at six points form four triangles. Prove that the circles circumscribed around out these triangles have a common point.
2018 NZMOC Camp Selection Problems, 8
Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.
2014 Contests, 2
The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.
2020 Greece Team Selection Test, 2
Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.
2015 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent.
Liana Topan
Indonesia MO Shortlist - geometry, g4
Given an acute triangle $ABC$ with $AB <AC$. Points $P$ and $Q$ lie on the angle bisector of $\angle BAC$ so that $BP$ and $CQ$ are perpendicular on that angle bisector. Suppose that point $E, F$ lie respectively at sides $AB$ and $AC$ respectively, in such a way that $AEPF$ is a kite. Prove that the lines $BC, PF$, and $QE$ intersect at one point.
Mathley 2014-15, 3
Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$ K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent.
Michel Bataille , France
2010 QEDMO 7th, 7
Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$ touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles $y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment $AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment $BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and $z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.
2009 QEDMO 6th, 11
The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.
2021 239 Open Mathematical Olympiad, 4
Symedians of an acute-angled non-isosceles triangle $ABC$ intersect at a point at point $L$, and $AA_1$, $BB_1$ and $CC_1$ are its altitudes. Prove that you can construct equilateral triangles $A_1B_1C'$, $B_1C_1A'$ and $C_1A_1B'$ not lying in the plane $ABC$, so that lines $AA' , BB'$ and $CC'$ and also perpendicular to the plane $ABC$ at point $L$ intersected at one point.
1961 Poland - Second Round, 2
Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.
1985 Tournament Of Towns, (080) T1
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
2017 Thailand TSTST, 1
In $\vartriangle ABC, D, E, F$ are the midpoints of $AB, BC, CA$ respectively. Denote by $O_A, O_B, O_C$ the incenters of $\vartriangle ADF, \vartriangle BED, \vartriangle CFE$ respectively. Prove that $O_AE, O_BF, O_CD$ are concurrent.
2023 Iranian Geometry Olympiad, 4
Let $ABC$ be a triangle and $P$ be the midpoint of arc $BAC$ of circumcircle of triangle $ABC$ with orthocenter $H$. Let $Q, S$ be points such that $HAPQ$ and $SACQ$ are parallelograms. Let $T$ be the midpoint of $AQ$, and $R$ be the intersection point of the lines $SQ$ and $PB$. Prove that $AB$, $SH$ and $TR$ are concurrent.
[i]Proposed by Dominik Burek - Poland[/i]
2009 Federal Competition For Advanced Students, P1, 4
Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.
2002 Junior Balkan Team Selection Tests - Romania, 3
Let $C_1(O_1)$ and $ C_2(O_2)$ be two circles such that $C_1$ passes through $O_2$. Point $M$ lies on $C_1$ such that $M \notin O_1O_2$. The tangents from $M$ at $O_2$ meet again $C_1$ at $A$ and $B$. Prove that the tangents from $A$ and $B$ at $C_2$ - others than $MA$ and $MB$ - meet at a point located on $C_1$.
2020 Yasinsky Geometry Olympiad, 6
In an isosceles triangle $ABC, I$ is the center of the inscribed circle, $M_1$ is the midpoint of the side $BC, K_2, K_3$ are the points of contact of the inscribed circle of the triangle with segments $AC$ and $AB$, respectively. The point $P$ lies on the circumcircle of the triangle $BCI$, and the angle $M_1PI$ is right. Prove that the lines $BC, PI, K_2K_3$ intersect at one point.
(Mikhail Plotnikov)
2022 Sharygin Geometry Olympiad, 4
Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of acute angled triangle $ABC$. $A_2$ be the touching point of the incircle of triangle $AB_1C_1$ with $B_1C_1$, points $B_2$, $C_2$ be defined similarly. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ concur.
1968 Spain Mathematical Olympiad, 6
Check and justify , if in every tetrahedron are concurrent:
a) The perpendiculars to the faces at their circumcenters.
b) The perpendiculars to the faces at their orthocenters.
c) The perpendiculars to the faces at their incenters.
If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.
2020 Greece National Olympiad, 2
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.