Found problems: 288
1974 Chisinau City MO, 83
Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.
2014 Czech and Slovak Olympiad III A, 2
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$.
(Day 1, 2nd problem
authors: Michal Rolínek, Jaroslav Švrček)
2013 Sharygin Geometry Olympiad, 7
Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.
1961 IMO, 6
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
1981 All Soviet Union Mathematical Olympiad, 326
The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.
1999 Estonia National Olympiad, 5
Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.
2016 Sharygin Geometry Olympiad, P10
Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.
1960 IMO, 7
An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.
1966 IMO Longlists, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
2022 Belarusian National Olympiad, 11.4
On plane circles $\omega_1, \omega_2, \omega_3$ with centers $O_1,O_2,O_3$ are given such that $\omega_1$ is externally tangent $\omega_2$ and $\omega_3$ at points $P, Q$ respectively. On $\omega_1$ point $C$ is chosen arbitrarily. Line $CP$ intersects $\omega_2$ at $B$, line $CQ$ intersects $\omega_3$ at $A$. Point $O$ is the circumcenter of $ABC$.
Prove that the locus of points $O$ (when $C$ changes) is a circle, the center of which lies on the circumcircle of $O_1O_2O_3$
2014 Contests, 2
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$.
(Day 1, 2nd problem
authors: Michal Rolínek, Jaroslav Švrček)
1986 IMO Shortlist, 1
Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
Estonia Open Senior - geometry, 2000.1.3
In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.
1996 Romania National Olympiad, 4
In the triangle $ABC$ the incircle $J$ touches the sides $BC$, $CA$, $AB$ in $D$, $E$, $F$, respectively. The segments $(BE)$ and $(CF)$ intersect $J$ in $G,H$. If $B$ and $C$ are fixed points, find the loci of points $A, D, E, F, G, H$ if $GH \parallel BC$ and the loci of the same points if $BCHG$ is an inscriptible quadrilateral.
1979 Bundeswettbewerb Mathematik, 2
A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.
1978 Romania Team Selection Test, 2
Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.
1962 German National Olympiad, 5
Given a plane $P$ and two fixed points $A$ and $B$ that do not lie in this plane. Denote two points $A'$ and $B'$ on plane $P$ and $M ,N$ the midpoints of the segments $AA'$, $BB'$.
a) Determine the locus of the midpoint of the segment MN if the points are $A'$ and $B'$ move arbitrarily in plane $P$.
b) A circle $O$ is considered in the plane $P$. Determine the locus $L$ of the midpoint of the segment $MN$ if the points $A'$ and $B'$ lie on the circle $O$ or inside it .
c) $A'$ is assumed to be fixed on the circle $O$ or inside it and $B'$ is assumed to be movable inside it , except for $O$. Determine the locus of the point $B'$ such the above certain locus $L$ remains the same
.
Note: For b) and c) the following cases should be considered:
1. $A'$ and $B'$ are different,
2. $A'$ and $B'$ coincide.
1990 Tournament Of Towns, (268) 2
A semicircle $S$ is drawn on $AB$ as diameter. For an arbitrary point $C$ in $S$ ($C\ne A$,$ C \ne B$), squares are attached to sides $AC$ and $BC$ of triangle $ABC$ outside the triangle. Find the locus of the midpoint of the segment joining the centres of the squares as $C$ moves along $S$.
(J Tabov, Sofia)
1998 Belarus Team Selection Test, 1
Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$.
S.Shikh
1994 Spain Mathematical Olympiad, 2
Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$. Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.
Estonia Open Junior - geometry, 1999.1.2
Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.
1968 Spain Mathematical Olympiad, 5
Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.
1982 Bulgaria National Olympiad, Problem 6
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.
1986 All Soviet Union Mathematical Olympiad, 424
Two circumferences, with the distance $d$ between centres, intersect in points $P$ and $Q$ . Two lines are drawn through the point $A$ on the first circumference ($Q\ne A\ne P$) and points $P$ and $Q$ . They intersect the second circumference in the points $B$ and $C$ .
a) Prove that the radius of the circle, circumscribed around the triangle$ABC$ , equals $d$.
b) Describe the set of the new circle's centres, if thepoint $A$ moves along all the first circumference.
2017 Tuymaada Olympiad, 8
Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$.
(K. Tyschu)