Problems

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

Prove that $S_{ABCD}\le (AB\times BC+AD\times DC)/2$.

Prove that $\mathrm{\angle }ABC>{90}^{\circ }$ 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.