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 \(S_{ABCD} \leq (AB \times BC + AD \times DC)/2\).
Let \(ABC\) be a triangle, 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.