(a+b+c)(b+c-a)(c+a-b)(a+b-c) By Formula

Jasonbradley

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Jun 16, 2024

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The Formula for (a+b+c)(b+c-a)(c+a-b)(a+b-c)

The expression (a+b+c)(b+c-a)(c+a-b)(a+b-c) can be simplified using a clever algebraic manipulation. This formula is often encountered in various mathematical contexts, particularly in geometry and trigonometry.

Understanding the Formula

The formula for this expression is:

(a+b+c)(b+c-a)(c+a-b)(a+b-c) = 2(ab+ac+bc)^2 - (a^4 + b^4 + c^4)

Proof of the Formula

We can prove this formula using algebraic manipulation:

1. Expand the first two factors:

(a+b+c)(b+c-a) = (b+c)^2 - a^2 = b^2 + 2bc + c^2 - a^2

2. Expand the last two factors:

(c+a-b)(a+b-c) = (a+b)^2 - c^2 = a^2 + 2ab + b^2 - c^2

3. Multiply the results from step 1 and step 2:

(b^2 + 2bc + c^2 - a^2)(a^2 + 2ab + b^2 - c^2) = (2ab + 2ac + 2bc)^2 - (a^4 + b^4 + c^4)

4. Simplify:

(2ab + 2ac + 2bc)^2 - (a^4 + b^4 + c^4) = 2(ab+ac+bc)^2 - (a^4 + b^4 + c^4)

Applications

This formula has several applications, including:

• Finding the volume of a tetrahedron: The formula can be used to find the volume of a tetrahedron with edge lengths a, b, and c.
• Solving geometric problems: The formula can be helpful in solving problems involving triangles, quadrilaterals, and other geometric shapes.
• Trigonometric identities: The formula can be used to derive trigonometric identities.

Conclusion

The formula for (a+b+c)(b+c-a)(c+a-b)(a+b-c) is a useful tool for simplifying expressions and solving mathematical problems. It's important to understand the proof and applications of this formula to fully appreciate its significance.