Question

Докажите, что если каждая диагональ четырехугольника делит его периметр пополам, то он является параллелограммом. 

Answer 1

The Relationship Between Diagonals and Perimeter in a Quadrilateral

To prove that if each diagonal of a quadrilateral divides its perimeter in half, then the quadrilateral is a parallelogram, we can use the following steps:

1. Let's assume we have a quadrilateral ABCD, where AC and BD are the diagonals. 2. We know that the perimeter of a quadrilateral is the sum of its four sides. Let's denote the lengths of the sides as AB, BC, CD, and DA, and the perimeter as P. 3. According to the given condition, each diagonal divides the perimeter in half. Therefore, we can write the following equations: - AC = (AB + CD) / 2 - BD = (BC + DA) / 2 4. We can simplify these equations to: - 2AC = AB + CD - 2BD = BC + DA 5. Now, let's consider the opposite sides of the quadrilateral. In a parallelogram, opposite sides are equal in length. Therefore, we can write the following equations: - AB = CD - BC = DA 6. Substituting these equations into the previous ones, we get: - 2AC = AB + AB - 2BD = BC + BC 7. Simplifying further, we have: - 2AC = 2AB - 2BD = 2BC 8. Dividing both sides of each equation by 2, we obtain: - AC = AB - BD = BC 9. From these equations, we can conclude that opposite sides of the quadrilateral are equal in length, which is a property of a parallelogram. 10. Therefore, if each diagonal of a quadrilateral divides its perimeter in half, then the quadrilateral is a parallelogram.

In conclusion, if each diagonal of a quadrilateral divides its perimeter in half, then the quadrilateral is a parallelogram. This can be proven by assuming a quadrilateral ABCD, using the given condition that the diagonals divide the perimeter in half, and showing that the opposite sides of the quadrilateral are equal in length, which is a property of a parallelogram.