Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Using only straightedge and compass construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).
Construct the triangle ABC by the medians \(m_a, m_b\) and \(m_c\).
Construct a triangle with the side \(c\), median to side \(a\), \(m_a\), and median to side \(b\), \(m_b\).
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 \(S_{ABCD} \leq (AB \times BC + AD \times DC)/2\).
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 \(|R - r| < d < R + r\).