Problems

A polygon is drawn around a circle of radius $r$. Prove that its area is equal to $pr$, where $p$ is the semiperimeter of the polygon.

The point $E$ is located inside the parallelogram $ABCD$. Prove that $S_{ABE}+S_{CDE}=S_{BCE}+S_{ADE}$.

Let $G,F,H$ and $I$ be the midpoints of the sides $CD,DA,AB,BC$ of the square $ABCD$, whose area is equal to $S$. Find the area of the quadrilateral formed by the straight lines $BG,DH,AF,CE$.

The diagonals of the quadrilateral $ABCD$ intersect at the point $O$. Prove that $S_{AOB}=S_{COD}$ if and only if $BC\parallel AD$.

a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.

b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.

Prove that the bisectors of a triangle intersect at one point.

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.