Question

Докажите, что сумма квадратов диагоналей параллелограмма равна удвоенной сумме квадратов двух смежных сторон.

Answer 1

Обозначим за x,y смежные стороны. Проведём диагонали. Пусть ABCD - параллелограмм, O - точка пересечения диагоналей. Пусть AO=CO=a, BO=DO=b. Рассмотрим треугольники ABO и ADO. В них по теореме косинусов выразим x и y:

x²=a²+b²-2abcosA
y²=a²+b²+2abcosA, где cosA - угол между диагоналями. Косинусы смежных углов противоположны, cos(180-A)=-cosA.

Тогда x²+y²=2(a²+b²). 
Диагонали равны 2a и 2b, поэтому сумма квадратов диагоналей равна 4(a²+b²)=2(x²+y²), что и требовалось доказать.

Answer 2

Proving the Sum of Squares of Diagonals in a Parallelogram

To prove that the sum of the squares of the diagonals of a parallelogram is equal to the doubled sum of the squares of its adjacent sides, we can use the properties of a parallelogram and the Pythagorean theorem.

Given: Let's consider a parallelogram with diagonals AC and BD, and sides AB and BC.

To Prove: The sum of the squares of the diagonals, AC^2 + BD^2, is equal to twice the sum of the squares of the adjacent sides, 2(AB^2 + BC^2).

Proof:

1. Using the Pythagorean Theorem: - In triangle ABC, applying the Pythagorean theorem, we have: - AB^2 + BC^2 = AC^2 (1) . - In triangle BCD, applying the Pythagorean theorem, we have: - BC^2 + CD^2 = BD^2 (2) .

2. Sum of the Squares of the Diagonals: - Adding equations (1) and (2), we get: - AB^2 + 2BC^2 + CD^2 = AC^2 + BD^2.

3. Using the Properties of a Parallelogram: - In a parallelogram, opposite sides are equal in length. - Therefore, AB = CD and BC = AD.

4. Substituting the Equal Sides: - Substituting AB = CD and BC = AD into the previous equation, we get: - 2(AB^2 + BC^2) = AC^2 + BD^2.

5. Conclusion: - Thus, we have proved that the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its adjacent sides.

This completes the proof.