Found problems: 333
2015 Balkan MO Shortlist, G7
Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.
the 8th XMO, 1
As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]
Ukrainian From Tasks to Tasks - geometry, 2016.3
In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point.
[img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]
Kvant 2020, M2600
Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent.
Maxim Didin
Swiss NMO - geometry, 2015.8
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.
1991 All Soviet Union Mathematical Olympiad, 540
$ABCD$ 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 $BD$.
2015 Sharygin Geometry Olympiad, P13
Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.
Geometry Mathley 2011-12, 13.4
Let $P$ be an arbitrary point in the plane of triangle $ABC$. Lines $PA, PB, PC$ meets the perpendicular bisectors of $BC,CA,AB$ at $O_a,O_b,O_c$ respectively. Let $(O_a)$ be the circle with center $O_a$ passing through two points $B,C$, two circles $(O_b), (O_c)$ are defined in the same manner. Two circles $(O_b), (O_c)$ meets at $A_1$, distinct from $A$. Points $B_1,C_1$ are defined in the same manner. Let $Q$ be an arbitrary point in the plane of $ABC$ and $QB,QC$ meets $(O_c)$ and $(O_b)$ at $A_2,A_3$ distinct from $B,C$. Similarly, we have points $B_2,B_3,C_2,C_3$. Let $(K_a), (K_b), (K_c)$ be the circumcircles of triangles $A_1A_2A_3, B_1B_2B_3, C_1C_2C_3$. Prove that
(a) three circles $(K_a), (K_b), (K_c)$ have a common point.
(b) two triangles $K_aK_bK_c, ABC$ are similar.
Trần Quang Hùng
2013 Balkan MO Shortlist, G4
Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$. The extension of the segment $KB$ meets the circle $(c)$ at point $E$. From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$. Prove that the lines $BE, ST$ and $AC$ are concurrent.
2008 Greece Junior Math Olympiad, 4
Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that :
i) triangle $BDZ$ is isosceles
ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$
iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$
2005 Sharygin Geometry Olympiad, 10.6
Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.
2020 Grand Duchy of Lithuania, 3
The tangents of the circumcircle $\Omega$ of the triangle $ABC$ at points $B$ and $C$ intersect at point $P$. The perpendiculars drawn from point $P$ to lines $AB$ and $AC$ intersect at points$ D$ and $E$ respectively. Prove that the altitudes of the triangle $ADE$ intersect at the midpoint of the segment $BC$.
VI Soros Olympiad 1999 - 2000 (Russia), 9.2
Let $A_1,$ $B_1$, $C_1$ be the touchpoints of the circle inscribed in the acute triangle $ABC$ ($A_1$ is the touchpoint with the side $BC$, etc.). Let $A_2$, $B_2$, $C_2$ be the intersection points of the altitudes of triangles $AB_1C_1$, $A_1BC_1$ and $A_1B_1C$ respectively. Prove that the lines $A_1A_2$ and $B_1B_2$ and $C_1C_2$ intersect at one point.
1956 Moscow Mathematical Olympiad, 328
In a convex quadrilateral $ABCD$, consider quadrilateral $KLMN$ formed by the centers of mass of triangles $ABC, BCD, DBA, CDA$. Prove that the straight lines connecting the midpoints of the opposite sides of quadrilateral $ABCD$ meet at the same point as the straight lines connecting the midpoints of the opposite sides of $KLMN$.
1997 Chile National Olympiad, 5
Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.
Kyiv City MO Juniors 2003+ geometry, 2020.7.41
In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.
2022 Durer Math Competition Finals, 4
$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.
2020 Switzerland - Final Round, 2
Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.
1979 Bundeswettbewerb Mathematik, 2
The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.
2007 Sharygin Geometry Olympiad, 8
Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,
2004 German National Olympiad, 2
Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$) to $k_1$ passing through $A$ and $B$ by $a$ and $b$, respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.
2000 BAMO, 2
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
2005 Sharygin Geometry Olympiad, 11.1
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
2011 Oral Moscow Geometry Olympiad, 6
Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.
2013 Greece JBMO TST, 2
Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle.
a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$
b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.