Construct a triangle with the side \(a\), the side \(b\) and height to side \(a\), \(h_a\).
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.
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 \(|R - r| < d < R + r\).
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 \(a^2 + b^2 + c^2 < 2 (ab + bc + ca)\).