Found problems: 333
Geometry Mathley 2011-12, 14.4
Two triangles $ABC$ and $PQR$ have the same circumcircles. Let $E_a, E_b, E_c$ be the centers of the Euler circles of triangles $PBC, QCA, RAB$. Assume that $d_a$ is a line through $Ea$ parallel to $AP$, $d_b, d_c$ are defined in the same manner. Prove that three lines $d_a, d_b, d_c$ are concurrent.
Nguyễn Tiến Lâm, Trần Quang Hùng
2019 Canadian Mathematical Olympiad Qualification, 6
Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.
Geometry Mathley 2011-12, 1.4
Given are three circles $(O_1), (O_2), (O_3)$, pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let $(X_1)$ be the circle externally tangent to $(O_1)$ and internally tangent to the circles $(O_2), (O_3),$ circles $(X_2), (X_3)$ are defined in the same manner. Let $(Y_1)$ be the circle internally tangent to $(O_1)$ and externally tangent to the circles $(O_2), (O_3)$, the circles $(Y_2), (Y_3)$ are defined in the same way. Let $(Z_1), (Z_2)$ be two circles internally tangent to all three circles $(O_1), (O_2), (O_3)$. Prove that the four lines $X_1Y_1, X_2Y_2, X_3Y_3, Z_1Z_2$ are concurrent.
Nguyễn Văn Linh
2011 China Northern MO, 2
As shown in figure , the inscribed circle of $ABC$ is intersects $BC$, $CA$, $AB$ at points $D$, $E$, $F$, repectively, and $P$ is a point inside the inscribed circle. The line segments $PA$, $PB$ and $PC$ intersect respectively the inscribed circle at points $X$, $Y$ and $Z$. Prove that the three lines $XD$, $YE$ and $ZF$ have a common point.
[img]https://cdn.artofproblemsolving.com/attachments/e/9/bbfb0394b9db7aa5fb1e9a869134f0bca372c1.png[/img]
2022 Yasinsky Geometry Olympiad, 6
Let $\omega$ be the circumscribed circle of the triangle $ABC$, in which $AC< AB$, $K$ is the center of the arc $BAC$, $KW$ is the diameter of the circle $\omega$. The circle $\gamma$ is inscribed in the curvilinear triangle formed by the segments $BC$, $AB$ and the arc $AC$ of the circle $\omega$. It turned out that circle $\gamma$ also touches $KW$ at point $F$. Let $I$ be the center of the triangle $ABC$, $M$ is the midpoint of the smaller arc $AK$, and $T$ is the second intersection point of $MI$ with the circle $\omega$. Prove that lines $FI$, $TW$ and $BC$ intersect at one point.
(Mykhailo Sydorenko)
Geometry Mathley 2011-12, 10.2
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
2019 Balkan MO Shortlist, G9
Given semicircle $(c)$ with diameter $AB$ and center $O$. On the $(c)$ we take point $C$ such that the tangent at the $C$ intersects the line $AB$ at the point $E$. The perpendicular line from $C$ to $AB$ intersects the diameter $AB$ at the point $D$. On the $(c)$ we get the points $H,Z$ such that $CD = CH = CZ$. The line $HZ$ intersects the lines $CO,CD,AB$ at the points $S, I, K$ respectively and the parallel line from $I$ to the line $AB$ intersects the lines $CO,CK$ at the points $L,M$ respectively. We consider the circumcircle $(k)$ of the triangle $LMD$, which intersects again the lines $AB, CK$ at the points $P, U$ respectively. Let $(e_1), (e_2), (e_3)$ be the tangents of the $(k)$ at the points $L, M, P$ respectively and $R = (e_1) \cap (e_2)$, $X = (e_2) \cap (e_3)$, $T = (e_1) \cap (e_3)$. Prove that if $Q$ is the center of $(k)$, then the lines $RD, TU, XS$ pass through the same point, which lies in the line $IQ$.
Geometry Mathley 2011-12, 2.4
Let $ABC$ be a triangle inscribed in a circle of radius $O$. The angle bisectors $AD,BE,CF$ are concurrent at $I$. The points $M,N, P$ are respectively on $EF, FD$, and $DE$ such that $IM, IN, IP$ are perpendicular to $BC,CA,AB$ respectively. Prove that the three lines $AM,BN, CP$ are concurrent at a point on $OI$.
Nguyễn Minh Hà
2014 IMAC Arhimede, 2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
2015 Sharygin Geometry Olympiad, P19
Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.
2000 Bosnia and Herzegovina Team Selection Test, 6
It is given triangle $ABC$ such that $\angle ABC = 3 \angle CAB$. On side $AC$ there are two points $M$ and $N$ in order $A - N - M - C$ and $\angle CBM = \angle MBN = \angle NBA$. Let $L$ be an arbitrary point on side $BN$ and $K$ point on $BM$ such that $LK \mid \mid AC$. Prove that lines $AL$, $NK$ and $BC$ are concurrent
1950 Moscow Mathematical Olympiad, 175
a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$.
b) for $n=3$ the same problem.
2015 Sharygin Geometry Olympiad, 6
The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D' $ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD' $ concur.
(A. Zaslavsky)
2021 Sharygin Geometry Olympiad, 10-11.1
.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.
2007 Regional Olympiad of Mexico Center Zone, 2
Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.
Cono Sur Shortlist - geometry, 2021.G3
Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.
2017 Balkan MO Shortlist, G8
Given an acute triangle $ABC$ ($AC\ne AB$) and let $(C)$ be its circumcircle. The excircle $(C_1)$ corresponding to the vertex $A$, of center $I_a$, tangents to the side $BC$ at the point $D$ and to the extensions of the sides $AB,AC$ at the points $E,Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $(C_1)$, $H$ the orthocenter of the triangle $EDZ$ and $N$ the midpoint of segment $EZ$. The parallel line through the point $l_a$ to the line $HL$ meets the line $HI$ at the point $G$. Prove that the perpendicular line $(e)$ through the point $N$ to the line $BC$ and the parallel line $(\delta)$ through the point $G$ to the line $IL$ meet each other on the line $HI_a$.
2015 NZMOC Camp Selection Problems, 3
Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.
2022 IFYM, Sozopol, 2
Let $k$ be the circumcircle of the acute triangle $ABC$. Its inscribed circle touches sides $BC$, $CA$ and $AB$ at points $D, E$ and $F$ respectively. The line $ED$ intersects $k$ at the points $M$ and $N$, so that $E$ lies between $M$ and $D$. Let $K$ and $L$ be the second intersection points of the lines $NF$ and $MF$ respectively with $k$. Let $AK \cap BL = Q$. Prove that the lines $AL$, $BK$ and $QF$ intersect at a point.
VMEO III 2006, 10.1
Let $ABC$ be a triangle inscribed in a circle with center $O$. Let $A_1$ be a point on arc $BC$ that does not contain $ A$ such that the line perpendicular to $OA$ at $A_1$ intersects the lines $AB$ and $AC$ at two points and the line segment joining those two points has as midpoint $A_1$. Points $B_1$, $C_1$ are determined similarly. Prove that the lines $AA_1$, $BB_1$, $CC_1$ are concurrent.
2014 Israel National Olympiad, 3
Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.
Kyiv City MO Juniors Round2 2010+ geometry, 2016.7.3
In an acute triangle $ABC$, the bisector $AL$, the altitude $BH$, and the perpendicular bisector of the side $AB$ intersect at one point. Find the value of the angle $BAC$.
1998 Tournament Of Towns, 2
On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
(P Kozhevnikov)
2021 Sharygin Geometry Olympiad, 10-11.7
Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.
Estonia Open Junior - geometry, 2005.1.3
In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.