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.