Let ABCD be a parallelogram. Prove that AB^2 + BC^2 + CD^2 +DA^2 = AC^2 + BD^2

To prove that in a parallelogram ABCD, the equation \( AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 \) holds true, we can utilize the properties of parallelograms and some basic principles from geometry. Let's break it down step by step.

Understanding the Properties of a Parallelogram

A parallelogram is defined as a quadrilateral where opposite sides are equal in length and parallel. Therefore, in parallelogram ABCD, we have:

• AB = CD
• BC = DA

Additionally, the diagonals of a parallelogram bisect each other. This means that the midpoints of diagonals AC and BD are the same.

Setting Up the Proof

We can use the distance formula and coordinate geometry to establish the relationship. Let’s place the parallelogram in a coordinate system for easier calculations. Assume the following coordinates:

• A(0, 0)
• B(a, 0)
• C(a + b, c)
• D(b, c)

Here, \( a, b, \) and \( c \) are lengths represented along the x and y axes respectively.

Calculating the Lengths of the Sides

Now, we can calculate the lengths of each side:

• AB = √[(a - 0)² + (0 - 0)²] = a
• BC = √[(a + b - a)² + (c - 0)²] = √[b² + c²]
• CD = √[(b - (a + b))² + (c - c)²] = a
• DA = √[(0 - b)² + (0 - c)²] = √[b² + c²]

Calculating the Diagonal Lengths

Next, we find the lengths of the diagonals AC and BD:

• AC = √[(a + b - 0)² + (c - 0)²] = √[(a + b)² + c²]
• BD = √[(a - b)² + (0 - c)²] = √[(a - b)² + c²]

Substituting Back into the Equation

Now we can substitute these lengths back into our equation:

Left side:

AB² + BC² + CD² + DA² = a² + (b² + c²) + a² + (b² + c²)

Which simplifies to: 2a² + 2(b² + c²)

Right side:

AC² + BD² = [(a + b)² + c²] + [(a - b)² + c²]

Expanding these: (a² + 2ab + b² + c²) + (a² - 2ab + b² + c²)

This simplifies to: 2a² + 2b² + 2c²

Equating Both Sides

Now we can see that both sides equal:

2a² + 2(b² + c²) = 2a² + 2b² + 2c²

Since both sides are equal, we have successfully proven that:

AB² + BC² + CD² + DA² = AC² + BD²

Conclusion

This relationship is a direct result of the properties of parallelograms, particularly the equality of opposite sides and the characteristics of the diagonals. By applying coordinate geometry and basic algebra, we can arrive at a clear proof demonstrating this important property. This proof not only reinforces our understanding of parallelograms but also illustrates the beauty of geometric relationships!