Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.

Let \( L \), \( M \) and \( N \) be the midpoints on the sides \( BC \), \( AC \) and \( AB\) of a triangle \( ABC \). Points \( D \), \( E \) and \( F \) are taken on the circle circumscribed to the triangle \( LMN \) so that the segments \( LD \), \( ME \) and \( NF \) are diameters of said circumference. Prove that the area of the hexagon \( LENDMF \) is equal to half the area of the triangle \( ABC \)

Through point $P$, which lies on one of the sides of the triangle $ABC$, draw a line dividing its area in half.

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

$S$ is a point inside triangle $ABC$ such that the areas of the triangles $ABS$, $BCS$ and $CAS$ are all equal. Prove that $S$ is the centroid of $ABC$.

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral . (G . Galperin , A . Savin, Moscow)

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.

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 $CDP$.

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area? (Problem from Latvia)

Triangle $ABC$ is given. The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively. When $ABC$ is rotated $180$ degrees about point $I$, triangle $A'B'C'$ results. Lines $AD, B'C'$ meet at $U$, lines $BE, A'C'$ meet at $V$, and lines $CF, A'B'$ meet at $W$. Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively. Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively. Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively. Six (not necessarily convex) quadrilaterals were colored orange: \[AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2\] Six other quadrilaterals were colored green: \[AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1\] Prove that the sum of the green areas equals the sum of the orange areas.

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

In a convex pentagon $ABCDE$ the areas of the triangles $ABC, ABD, ACD$ and $ADE$ are all equal to the same value x. What is the area of the triangle $BCE$?

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]

$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.

Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$. [img]https://cdn.artofproblemsolving.com/attachments/1/7/b29bdbb1f2b103be1f3cb2650b3bfff352024a.png[/img]

If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one. (A. Savin, Moscow)

Determine all the quadrilaterals that can be divided by a diagonal into two triangles of equal area and equal perimeter.

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.