In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.

Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.

The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.

$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

Trapezoid $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines $AC, BC, AD$ and $BD$ is inscribed.

In the tetrahedron $DABC$ : $\angle ACB = \angle ADB$, $(CD) \perp (ABC)$. In triangle $ABC$, the altitude $h$ drawn to the side $AB$ and the distance $d$ from the center of the circumscribed circle to this side are given. Find the length of the $CD$.