Question

Доказать, что сумма квадратов диагоналей параллелограмма равна сумме квадратов всех его сторон.

Answer 1

Допустим, имеем параллелограмм ABCD, в котором AC и BD - диагонали.
Доказательство:
1. Необходимо опустить перпендикуляры BK и CF на прямую, которая содержит сторону AD. 
2. Рассмотрим ΔBDK:
По теореме Пифагора:
BD²=KD²+BK²
3. Рассмотрим ΔACF:
По теореме Пифагора:
AC²=AF²+CF²
4. Складываем два выражения в столбик:
BD²=KD²+BK² 
+
AC²=AF²+CF²
=
AC²+BD²=KD²+BK²+AF²+CF²
По свойству высот в параллелограмме, BK=CF ⇒ AC²+BD²=2BK²+KD²+AF²
5. Рассмотрим ΔABK:
По теореме Пифагора:
BK²=AB²-AK²
6. Так как KD=AD-AK, AF=AD+FD ⇒ AC²+BD²=2(AB²-AK²)+(AD-AK)²+(AD+FD)²
7. BK=CF, AB=CD ⇒ ΔABK=ΔDCF - по свойству катета и гипотенузы ⇒ AK=DF ⇒ 
AC²+BD²=2(AB²-AK²)+(AD-AK)²+(AD+AK)²
AC²+BD²=2AB²-2AK²+AD²-2AD*AK+AK²+AD²+2AD*AK+AK²
AC²+BD²=2AB²+2AD²
AC²+BD²=2(AB²+AD²)
Что и требовалось доказать.

Answer 2

Proof that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides

To prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides, we can use the properties of parallelograms and basic geometry principles.

Let's consider a parallelogram ABCD, where AB and CD are the parallel sides, and AC and BD are the diagonals.

To begin, we can use the fact that opposite sides of a parallelogram are congruent. This means that AB = CD and AD = BC.

Next, we can use the fact that opposite angles of a parallelogram are congruent. This means that angle A = angle C and angle B = angle D.

Using these properties, we can divide the parallelogram into two congruent triangles: triangle ABC and triangle CDA.

Now, let's calculate the sum of the squares of the diagonals.

The length of diagonal AC can be calculated using the Pythagorean theorem in triangle ABC. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In triangle ABC, we have: - AB as one side - BC as another side - AC as the hypotenuse

Using the Pythagorean theorem, we can write: AC^2 = AB^2 + BC^2

Similarly, the length of diagonal BD can be calculated using the Pythagorean theorem in triangle CDA: BD^2 = CD^2 + AD^2

Now, let's calculate the sum of the squares of all the sides of the parallelogram.

The sum of the squares of all the sides can be calculated by adding the squares of each side.

Sum of squares of all sides = AB^2 + BC^2 + CD^2 + AD^2

To prove that the sum of the squares of the diagonals is equal to the sum of the squares of all the sides, we need to show that AC^2 + BD^2 is equal to the sum of squares of all sides.

Let's substitute the values of AC^2 and BD^2 using the Pythagorean theorem:

AC^2 = AB^2 + BC^2 BD^2 = CD^2 + AD^2

Now, let's add these two equations together:

AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + AD^2

We can see that the sum of the squares of the diagonals (AC^2 + BD^2) is equal to the sum of the squares of all the sides (AB^2 + BC^2 + CD^2 + AD^2).

Therefore, we have proved that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides.

Note: The sources provided did not contain specific information related to this proof. However, the proof is based on well-known properties of parallelograms and the Pythagorean theorem, which are fundamental concepts in geometry.