Found problems: 288
2018 IMAR Test, 1
Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$.
[i]* * *[/i]
2010 Sharygin Geometry Olympiad, 6
An arbitrary line passing through vertex $B$ of triangle $ABC$ meets side $AC$ at point $K$ and the circumcircle in point $M$. Find the locus of circumcenters of triangles $AMK$.
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.
1963 IMO Shortlist, 2
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.
2005 Sharygin Geometry Olympiad, 5
There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles:
a) points of intersection of heights,
b) the intersection points of the medians,
c) the centers of the circumscribed circles.
1992 IMO Shortlist, 20
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 $\,PQR$.
2011 IMAR Test, 1
Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.
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.
1988 Greece National Olympiad, 2
Given regular $1987$ -gon on plane with vertices $A_1, A_2,..., A_{1987}$. Find locus of points M of the plane sych that $$\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987$$.
1997 Argentina National Olympiad, 5
Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.
2018 Oral Moscow Geometry Olympiad, 5
The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.
1948 Putnam, B2
A circle moves so that it is continually in the contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.
2023 Sharygin Geometry Olympiad, 19
A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$.
1949-56 Chisinau City MO, 61
Find the locus of the projections of a given point on all planes containing another point $B$.
1986 IMO, 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 $X$.
1980 Spain Mathematical Olympiad, 7
The point $M$ varies on the segment $AB$ that measures $2$ m.
a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ .
b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation).
c) Find the area of the enclosure between the curve obtained and the coordinate axes.
1963 All Russian Mathematical Olympiad, 040
Given an isosceles triangle. Find the set of the points inside the triangle such, that the distance from that point to the base equals to the geometric mean of the distances to the sides.
VMEO III 2006, 11.2
Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.
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)
2006 Sharygin Geometry Olympiad, 14
Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.
2021 Sharygin Geometry Olympiad, 10-11.8
On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_2$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.
Ukraine Correspondence MO - geometry, 2006.3
Find the locus of the points of intersection of the altitudes of the triangles inscribed in a given circle.
1996 Abels Math Contest (Norwegian MO), 1
Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point.
A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$.
Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.
2000 Tournament Of Towns, 3
$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle.
(M Panov)
2017 Oral Moscow Geometry Olympiad, 6
Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.