Problems

Prove that a convex quadrilateral $ABCD$ can be drawn inside a circle if and only if $\mathrm{\angle }ABC+\mathrm{\angle }CDA={180}^{\circ }$.

Prove that a convex quadrilateral $ICEF$ can contain a circle if and only if $IC+EH=CE+IF$.

a) Prove that the axes of symmetry of a regular polygon intersect at one point.

b) Prove that the regular $2n$-gon has a centre of symmetry.

a) The convex $n$-gon is cut by diagonals that do not cross to form triangles. Prove that the number of these triangles is equal to $n-2$.

b) Prove that the sum of the angles at the vertices of a convex $n$-gon is $(n-2)\times {180}^{\circ }$.

The triangle $ABC$ is given. Find the locus of the point $X$ satisfying the inequalities $AX\le CX\le BX$.

Find the locus of the points $X$ such that the tangents drawn from $X$ to a given circle have a given length.

The point $A$ is fixed on a circle. Find the locus of the point $X$ which divides the chords that end at point $A$ in a $1:2$ ratio, starting from the point $A$.

Let $O$ be the center of the rectangle $ABCD$. Find the geometric points of $M$ for which $AM\ge OM,BM\ge OM$, $CM\ge OM$, and $DM\ge OM$.

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.