Found problems: 116
2014 Junior Balkan Team Selection Tests - Romania, 4
In a circle, consider two chords $[AB], [CD]$ that intersect at $E$, lines $AC$ and $BD$ meet at $F$. Let $G$ be the projection of $E$ onto $AC$. We denote by $M,N,K$ the midpoints of the segment lines $[EF] ,[EA]$ and $[AD]$, respectively. Prove that the points $M, N,K,G$ are concyclic.
1989 Tournament Of Towns, (234) 2
Three points $K, L$ and $M$ are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.
2004 Oral Moscow Geometry Olympiad, 1
$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.
2013 JBMO Shortlist, 6
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.
2014 Canadian Mathematical Olympiad Qualification, 6
Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.
2008 Oral Moscow Geometry Olympiad, 5
Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides.
(A. Zaslavsky)
Ukraine Correspondence MO - geometry, 2015.11
Let $ABC$ be an non- isosceles triangle, $H_a$, $H_b$, and $H_c$ be the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively, and $M_a$, $M_b$, and $M_c$ be the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The circumscribed circles of triangles $AH_bH_c$ and $AM_bM_c$ intersect for second time at point $A'$. The circumscribed circles of triangles $BH_cH_a$ and $BM_cM_a$ intersect for second time at point $B'$. The circumscribed circles of triangles $CH_aH_b$ and $CM_aM_b$ intersect for second time at point $C'$. Prove that points $A', B'$ and $C'$ lie on the same line.
2015 BAMO, 4
In a quadrilateral, the two segments connecting the midpoints of its opposite sides are equal in length. Prove that the diagonals of the quadrilateral are perpendicular.
(In other words, let $M,N,P,$ and $Q$ be the midpoints of sides $AB,BC,CD,$ and $DA$ in quadrilateral $ABCD$. It is known that segments $MP$ and $NQ$ are equal in length. Prove that $AC$ and $BD$ are perpendicular.)
1996 Greece Junior Math Olympiad, 2
In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that:
a) $AK=3CK$
b) $HK=3HG$
c) $BE=3EI$
d) $(EGH)=\frac{1}{32}(ABC)$
Notation $(...)$ stands for area of $....$
2016 Rioplatense Mathematical Olympiad, Level 3, 3
Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.
2005 Junior Tuymaada Olympiad, 7
The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $
1998 Belarus Team Selection Test, 1
Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$.
Ya. Konstantinovski
2022 Bolivia IMO TST, P3
On $\triangle ABC$, let $M$ the midpoint of $AB$ and $N$ the midpoint of $CM$. Let $X$ a point such that $\angle XMC=\angle MBC$ and $\angle XCM=\angle MCB$ with $X,B$ in opposite sides of line $CM$. Let $\Omega$ the circumcircle of triangle $\triangle AMX$
[b]a)[/b] Show that $CM$ is tangent to $\Omega$
[b]b)[/b] Show that the lines $NX$ and $AC$ meet at $\Omega$
Kyiv City MO Juniors 2003+ geometry, 2018.7.41
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
2015 Iran Geometry Olympiad, 2
In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$.
by Davood Vakili
1949-56 Chisinau City MO, 32
Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.
Estonia Open Junior - geometry, 2014.1.5
In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.
2010 Junior Balkan Team Selection Tests - Romania, 2
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$
2011 Peru MO (ONEM), 3
Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.
2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.
2008 Estonia Team Selection Test, 5
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
1999 Tournament Of Towns, 4
Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$
(a) divides the perimeter of triangle $ABC$ in half,
(b) is parallel to the bisector of angle $ACB$.
( L Emelianov)
2016 Hanoi Open Mathematics Competitions, 13
Let $H$ be orthocenter of the triangle $ABC$. Let $d_1, d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB, AC$ at $D, E$ and the line d_2 intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$ if and only if $F$ is the midpoint of segment $BC$.
2005 Sharygin Geometry Olympiad, 9.3
Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.
2014 Indonesia MO Shortlist, G5
Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.