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 $CD$.

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.

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 $X$.

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.

Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies. [i]Proposed by T. Vitanov, E. Kolev[/i]

Given the segment $ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

Rays $l_1$ and $l_2$ pass through a point $O$. Segments $OA_1$ and $OB_1$ on $l_1$, and $OA_2$ and $OB_2$ on $l_2$, are drawn so that $\frac{OA_1}{OA_2} \ne \frac{OB_1}{OB_2}$ . Find the set of all intersection points of lines $A_1A_2$ and $B_1B_2$ as $l_2$ rotates around $O$ while $l_1$ is fixed.

Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

Determine the locus of points, from which the tangent segments to two given circles are equal.

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.

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

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 $X$.

Given two skew perpendicular lines in space, find the set of the midpoints of all segments of given length with the endpoints on these lines.

Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.

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]

Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ drawn from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?

Let $S$ be the circle of center $O$ and radius $R$, and let $A, A'$ be two diametrically opposite points in $S$. Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$, different from $AA '$ and from the perpendicular on $AA '$. Let $B$ and $C$ be the intersection points of $\ell$ with $S$ and let $M$ be the midpoint of $BC$. a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the intersection point of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies . b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .

Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.

Given a circle $\gamma$ and two points $A$ and $B$ in its plane. By $B$ passes a variable secant that intersects $\gamma$ at two points $M$ and $N$. Determine the locus of the centers of the circles circumscribed to the triangle $AMN$.

Let $ABC$ an acute-angle triangle. Let $R$ be a rectangle with vertices in the edges of $ABC$. Let $O$ be the center of $R$. a) Find the locus of all the points $O$. b) Decide if there is a point that is the center of three of these rectangles.

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$. Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$