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 \((n - 2) \times 180^{\circ}\).
Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).
Construct a straight line passing through a given point and tangent to a given circle.
Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).
Construct the triangle ABC by the medians \(m_a, m_b\) and \(m_c\).
Construct a triangle with the side \(c\), median to side \(a\), \(m_a\), and median to side \(b\), \(m_b\).
Construct a triangle with the side \(a\), the side \(b\) and height to side \(a\), \(h_a\).