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

**. Let points**

*ABCD***,**

*A'***,**

*B'***and**

*C'***be midpoints of sides**

*D'***,**

*CD***,**

*DA***and**

*AB***correspondingly. Prove that segments**

*BC***,**

*AA'***,**

*BB'***and**

*CC'***form a square with each side having a length of 2/5 of the length of each of these segments.**

*DD'*8. Given a square

**. Let points**

*ABCD***,**

*A'***,**

*B'***and**

*C'***be positioned on sides**

*D'***,**

*AB***,**

*BC***and**

*CD***correspondingly, such that segments**

*DA***,**

*AA'***,**

*BB'***and**

*CC'***are congruent. Prove that**

*DD'***is a square.**

*A'B'C'D'*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.

## No comments:

Post a Comment