Found problems: 67
Let $ABCD$ be a convex quadrilateral. Determine all points $M$, which lie inside $ABCD$, such that the areas of $ABCM$ and $AMCD$ are equal.
$S$ 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 $ABC$.
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)
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
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.
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 $MA$.
In the plane, a triangle is given. Determine all points $P$ in the plane such that each line through $P$ that divides the triangle into two parts with the same area must pass through one of the vertices of the triangle.
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal 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 an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.
A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.
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 \( ABC \)
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 $BCE$?
In this exercise we only consider convex quadrilaterals.
(a) For such a quadrilateral $ABCD$, determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas.
(b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$.
(c) Find out which quadrilaterals $ABCD$ contains a point $P$, so that the triangles $PAB$, $PBC$, $PCD$ and $PDA$ have equal areas.
Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .