Problems

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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 SABE+SCDE=SBCE+SADE.

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 SAOB=SCOD if and only if BCAD.

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 ABC+CDA=180.

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.