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 \(AC > BD\).
Prove that if two opposite angles of a quadrilateral are obtuse, then the diagonal connecting the vertices of these angles is shorter than the other diagonal.
Prove that the sum of the distances from an arbitrary point to three vertices of an isosceles trapezium is greater than the distance from this point to the fourth vertex.
Prove that if the angles of a convex pentagon form an arithmetic progression, then each of them is greater than \(36^{\circ}\).
On a line segment of length 1, \(n\) points are given. Prove that the sum of the distances from some point out of the ones on the segment to these points is no less than \(n / 2\).