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Let a, b, and c be the side lengths of a triangle. If | a² b² c² | | (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 is isosceles.

Solution

The determinant equals 4(a-b)(a-c)(b-c). It is zero precisely when at least two of a,b,c are equal, i.e. the triangle is isosceles (which also includes the equilateral case, so 'cannot be equilateral' is false).

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