Problems

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 $|R-r|<d<R+r$.

A triangle of area 1 with sides $a\le b\le c$ is given. Prove that $b\ge \sqrt{2}$.

In the quadrilateral $ABCD$, the angles $A$ and $B$ are equal, and $\mathrm{\angle }D>\mathrm{\angle }C$. Prove that $AD<BC$.

In the trapezoid $ABCD$, the angles at the base $AD$ satisfy the inequalities $\mathrm{\angle }A<\mathrm{\angle }D<{90}^{\circ }$. Prove that $AC>BD$.