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 \(\angle CDE = 90^{\circ}\) 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.