Found problems: 288
Cono Sur Shortlist - geometry, 2003.G6
Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$ with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.
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)
1953 Czech and Slovak Olympiad III A, 4
Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.
1957 Moscow Mathematical Olympiad, 351
Given two concentric circles and a pair of parallel lines. Find the locus of the fourth vertices of all rectangles with three vertices on the concentric circles, two vertices on one circle and the third on the other and with sides parallel to the given lines.
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 $a$.
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.
Ukrainian TYM Qualifying - geometry, 2015.23
An acute-angled triangle $ABC$ is given, through the vertices $B$ and $C$ of which a circle $\Omega$, $A \notin \Omega$, is drawn. We consider all points $P \in \Omega$, that do not lie on none of the lines $AB$ and $AC$ and for which the common tangents of the circumscribed circles of triangles $APB$ and $APC$ are not parallel. Let $X_P$ be the point of intersection of such two common tangents.
a) Prove that the locus of points $X_P$ lies to some two lines.
b) Prove that if the circle $\Omega$ passes through the orthocenter of the triangle $ABC$, then one of these lines is the line $BC$.
2016 Tournament Of Towns, 2
On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.
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.
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 $\,PQR$.
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.
2004 Tournament Of Towns, 4
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
1945 Moscow Mathematical Olympiad, 098
A right triangle $ABC$ moves along the plane so that the vertices $B$ and $C$ of the triangle’s acute angles slide along the sides of a given right angle. Prove that point $A$ fills in a line segment and find its length.
1964 Bulgaria National Olympiad, Problem 3
There are given two intersecting lines $g_1,g_2$ and a point $P$ in their plane such that $\angle(g1,g2)\ne90^\circ$. Its symmetrical points on any point $M$ in the same plane with respect to the given lines are $M_1$ and $M_2$. Prove that:
(a) the locus of the point $M$ for which the points $M_1,M_2$ and $P$ lie on a common line is a circle $k$ passing through the intersection point of $g_1$ and $g_2$.
(b) the point $P$ is an orthocenter of a triangle, inscribed in the circle $k$ whose sides lie at the lines $g_1$ and $g_2$.
II Soros Olympiad 1995 - 96 (Russia), 9.9
Two points $A$ and $B$ are given on the plane. An arbitrary circle passes through $B$ and intersects the straight line $AB$ for second time at a point $K$, different from $A$. A circle passing through $A$, $K$ and the center of the first circle intersects the first one for second time at point $M$. Find the locus of points $M$.
1960 IMO Shortlist, 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.
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.
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)
Kyiv City MO Seniors 2003+ geometry, 2007.10.3
The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$.
a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$.
b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.
1962 All-Soviet Union Olympiad, 2
Given a fixed circle $C$ and a line L through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle with center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?
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.
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.
2000 Austrian-Polish Competition, 7
Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that:
(i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely,
(ii) The triangles $ABC$ and $A_0B_0C_0$ are similar.
Find the possible positions of the circumcenter of triangle $ABC$.
1980 All Soviet Union Mathematical Olympiad, 287
The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.
1971 Czech and Slovak Olympiad III A, 5
Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$