Problems

Construct a circle with a given centre, tangent to a given circle.

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 $A^{\prime }{B}^{\prime }$ are given on a plane. Construct the point $O$ so that the triangles $AOB$ and $A^{\prime }O{B}^{\prime }$ 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 $S_{ABC}\le AB\times BC/2$.