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Let a, b, and c be the side lengths of a triangle. If | a^2 b^2 c^2 | | (a+1)^2 (b+1)^2 (c+1)^2 | | (a−1)^2 (b−1)^2 (c−1)^2 | = 0, then which of the following must be true?

1. The triangle ABC cannot be equilateral.
2. The triangle ABC is a right-angled isosceles triangle.
3. The triangle ABC is isosceles.
4. None of these.

Correct answer: The triangle ABC cannot be equilateral.

Solution

The determinant being zero indicates that the rows of the matrix are linearly dependent, which implies that the side lengths a, b, and c cannot all be equal, thus ruling out the possibility of the triangle being equilateral.

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