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In a tetrahedron OABC (with O as origin), opposite edges are equal: |OA| = |BC| = a, |OB| = |AC| = b, and |OC| = |AB| = c. Let G1 be the centroid of face ABC and G2 be the centroid of face AOC. If OG1 is perpendicular to BG2, find the value of (a² + c²)/b².

1. 2
2. 3
3. 6
4. 9

Correct answer: 3

Solution

Using the standard isosceles tetrahedron coordinates, the perpendicularity condition OG1 * BG2 = 0 yields q² = p² + r². Substituting into a² = 4(q²+r²), b² = 4(p²+r²), c² = 4(p²+q²) gives (a²+c²)/b² = 3(p²+r²)/(p²+r²) = 3.

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