 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2010/b/4/c/solution

Let

u = (1, 1, 1)

and

v = (1, α, β)

. The vector representing

r_{23}

satisfies

(α β - 1, α (β - 1), β (α - 1)) = α β u - v .

Consequently the three representative vectors are linearly dependent, so $r_{12}, r_{13}, r_{23}$ are collinear in the projective plane. Equivalently their three-by-three determinant vanishes. This is the two-line configuration of Pappus's hexagon theorem.

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