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 \(|R - r| < d < R + r\).
Prove that \((a + b - c)/2 < m_c < (a + b)/2\), where \(a\), \(b\) and \(c\) are the lengths of the sides of an arbitrary triangle and \(m_c\) is the median to side \(c\).
\(a\), \(b\) and \(c\) are the lengths of the sides of an arbitrary triangle. Prove that \(a = y + z\), \(b = x + z\) and \(c = x + y\), where \(x\), \(y\) and \(z\) are positive numbers.
a, b and c are the lengths of the sides of an arbitrary triangle. Prove that \(a^2 + b^2 + c^2 < 2 (ab + bc + ca)\).
In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.
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 \(AC > BD\).