Problems in Geometry Series 3

I have uploaded a new lecture to http://www.unizor.com/ with the following geometrical problems.

1. Prove that in an equilateral triangle sum of distances of any internal point to all three sides is constant and is equal to an altitude of a triangle.

2. Prove that a parallelogram with congruent diagonals is a rectangle.

3. Prove that a parallelogram with perpendicular diagonals is a rhombus.

4. Prove that a parallelogram with a diagonal being an angle bisector is a rhombus.

5. Given a rhombus. From a point of intersection of its diagonals we dropped perpendiculars to all four sides. Prove that points of intersection of these perpendiculars with sides form a rectangle.

6. Prove that angle bisectors of a rectangle form a square.

7. Given a square ABCD. Let points A', B', C' and D' be midpoints of sides CD, DA, AB and BC correspondingly. Prove that segments AA', BB', CC' and DD' form a square with each side having a length of 2/5 of the length of each of these segments.

8. Given a square ABCD. Let points A', B', C' and D' be positioned on sides AB, BC, CD and DA correspondingly, such that segments AA', BB', CC' and DD' are congruent. Prove that A'B'C'D' is a square.

9. What condition should a quadrangle satisfy, if a new quadrangle with vertices at midpoints of each its side form
(a) parallelogram,
(b) rhombus,
(c) rectangle,
(d) square.