Found problems: 288
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.
1977 Bulgaria National Olympiad, Problem 4
Vertices $A$ and $C$ of the quadrilateral $ABCD$ are fixed points of the circle $k$ and each of the vertices $B$ and $D$ is moving to one of the arcs of $k$ with ends $A$ and $C$ in such a way that $BC=CD$. Let $M$ be the intersection point of $AC$ and $BD$ and $F$ is the center of the circumscribed circle around $\triangle ABM$. Prove that the locus of $F$ is an arc of a circle.
[i]J. Tabov[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.5
An isosceles triangle $ABC$ ($AB = BC$) is given on the plane. Find the locus of points $M$ of the plane such that $ABCM$ is a convex quadrilateral and $\angle MAC + \angle CMB = 90^o$.
1967 IMO Shortlist, 3
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
2009 Switzerland - Final Round, 1
Let $P$ be a regular hexagon. For a point $A$, let $d_1\le d_2\le ...\le d_6$ the distances from $A$ to the six vertices of $P$, ordered by magnitude. Find the locus of all points $A$ in the interior or on the boundary of $P$ such that:
(a) $d_3$ takes the smallest possible value.
(b) $d_4$ takes the smallest possible value.
2017 Vietnamese Southern Summer School contest, Problem 4
Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$.
1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$.
2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.
1954 Czech and Slovak Olympiad III A, 4
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
1924 Eotvos Mathematical Competition, 2
If $O$ is a given point, $\ell$ a given line, and $a$ a given positive number, find the locus of points $P$ for which the sum of the distances from $P$ to $O$ and from $P$ to $\ell$ is $a$.
2021 Stars of Mathematics, 3
Let $ABC$ be a triangle, let its $A$-symmedian cross the circle $ABC$ again at $D$, and let $Q$ and $R$ be the feet of the perpendiculars from $D$ on the lines $AC$ and $AB$, respectively. Consider a variable point $X$ on the line $QR$, different from both $Q$ and $R$. The line through $X$ and perpendicular to $DX$ crosses the lines $AC$ and $AB$ at $V$ and $W$, respectively. Determine the geometric locus of the midpoint of the segment $VW$.
[i]Adapted from American Mathematical Monthly[/i]
2011 Tournament of Towns, 3
Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?
1941 Moscow Mathematical Olympiad, 074
A point $P$ lies outside a circle. Consider all possible lines drawn through $P$ so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.
2013 IFYM, Sozopol, 2
The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.
2010 Laurențiu Panaitopol, Tulcea, 3
Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $
[i]Laurențiu Panaitopol[/i]
1986 IMO Shortlist, 16
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 $X$.
1970 Czech and Slovak Olympiad III A, 5
Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$
1955 Moscow Mathematical Olympiad, 289
Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.
2018 Peru Cono Sur TST, 7
Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$. The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$. Let $T$ be the intersection point of lines $LA$ and $ND$. Find the locus of $T$ as $K$ varies along segment $AD$.
1969 IMO Longlists, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
1987 Tournament Of Towns, (148) 5
Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle .
(J . Tabov , Sofia)
1992 IMO, 1
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
2006 Sharygin Geometry Olympiad, 22
Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.
2015 Sharygin Geometry Olympiad, P20
Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.
1965 Vietnam National Olympiad, 2
$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.
Find the locus of $X$.
What happens to $X$ as $M$ tends to
(1) $D$,
(2) $C$?
Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.
1987 Tournament Of Towns, (151) 2
Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .
Ukrainian TYM Qualifying - geometry, 2013.6
Given a circle $\omega$, on which marks the points $A,B,C$. Let $BF$ and $CE$ be the altitudes of the triangle $ABC$, $M$ be the midpoint of the side $AC$. Find a the locus of the intersection points of the lines $BF$ and E$M$ for all positions of point $A$ , as $A$ moves on $\omega$.