The point \(E\) is located inside the parallelogram \(ABCD\). Prove that \(S_{ABE} + S_{CDE} = S_{BCE} + S_{ADE}\).
Let \(G,F,H\) and \(I\) be the midpoints of the sides \(CD, DA, AB, BC\) of the square \(ABCD\), whose area is equal to \(S\). Find the area of the quadrilateral formed by the straight lines \(BG,DH,AF,CE\).
The diagonals of the quadrilateral \(ABCD\) intersect at the point \(O\). Prove that \(S_{AOB} = S_{COD}\) if and only if \(BC \parallel AD\).
a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.
b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.
Prove that the bisectors of a triangle intersect at one point.
A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.
Prove that a convex quadrilateral \(ABCD\) can be drawn inside a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).
a) Prove that the axes of symmetry of a regular polygon intersect at one point.
b) Prove that the regular \(2n\)-gon has a centre of symmetry.
a) The convex \(n\)-gon is cut by diagonals that do not cross to form triangles. Prove that the number of these triangles is equal to \(n - 2\).
b) Prove that the sum of the angles at the vertices of a convex \(n\)-gon is \((n - 2) \times 180^{\circ}\).