Problems

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle $CDE$ the angle $\mathrm{\angle }CDE={90}^{\circ }$ and the line $DH$ is the median. A circle with center $A$ is inscribed in the triangle $CDH$ and is tangent to the segment $DH$ in its middle, let’s denote it as $G$, so $GH=DG$. Find the angles of the triangle $CDE$.

Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle $EFG$ the line $EH$ is the median. Two circles with centres $A$ and $C$ are inscribed into triangles $EFH$ and $EGH$ respectively, they are tangent to the median $EH$ at the points $B$ and $D$. Find the length of $BD$ if $EF-EG=2$.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a square, or slightly harder in a shape of a given rectangle.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a parallelogram.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a shape like this

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a given triangle.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a hexagon

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a circle.

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of a young moon (see the picture below)

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a ring.