Problems

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 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 $S_{ABC}\le AB\times BC/2$.

Construct the triangle $ABC$ along the side $a$, the height $h_a$ and the angle $A$.

Construct a right-angled triangle along the leg and the hypotenuse.

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