Problems

A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.

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 $(n-2)\times {180}^{\circ }$.

a) Find the locus of the points that are equidistant from two parallel lines.

b) Find the locus of the points that are equidistant from two intersecting lines.

Find the locus of the midpoints of the segments, the ends of which are found on two given parallel lines.

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 $A$.