Inside an angle two points, \(A\) and \(B\), are given. Construct a circle which passes through these points and cuts the sides of the angle into equal segments.
Two segments \(AB\) and \(A'B'\) are given on a plane. Construct the point \(O\) so that the triangles \(AOB\) and \(A'OB'\) are similar (the same letters denote the corresponding vertices of similar triangles).
Using a right angle, draw a straight line through the point \(A\) parallel to the given line \(l\).
Prove that \(S_{ABC} \leq AB \times BC/2\).
Prove that \(S_{ABCD} \leq (AB \times BC + AD \times DC)/2\).
Prove that \(\angle ABC > 90^{\circ}\) if and only if the point \(B\) lies inside a circle with diameter \(AC\).
The radii of two circles are \(R\) and \(r\), and the distance between their centres is equal \(d\). Prove that these circles intersect if and only if \(|R - r| < d < R + r\).
A triangle of area 1 with sides \(a \leq b \leq c\) is given. Prove that \(b \geq \sqrt{2}\).
In the quadrilateral \(ABCD\), the angles \(A\) and \(B\) are equal, and \(\angle D > \angle C\). Prove that \(AD < BC\).
In the trapezoid \(ABCD\), the angles at the base \(AD\) satisfy the inequalities \(\angle A < \angle D < 90^{\circ}\). Prove that \(AC > BD\).