It is easy to construct one equilateral triangle from three identical matches. Can we make four equilateral triangles by adding just three more matches identical to the original ones?
Construct a straight line passing through a given point and tangent to a given circle.
Three segments whose lengths are equal to \(a, b\) and \(c\) are given. 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.
Two segments \(AB\) and \(A'B'\) are given on a plane. Construct the point \(O\) so that the triangles \(AOB\) and \(A'OB'\) are similar (the same letters denote the corresponding vertices of similar triangles).
Using a right angle, draw a straight line through the point \(A\) parallel to the given line \(l\).
Prove that \(S_{ABC} \leq AB \times BC/2\).