 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/12/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The relation rearranges to

(z_{1} + z_{3}) /2 = (z_{2} + z_{4}) /2

A C

and

B D

have the same midpoint. Equivalently

z_{2} - z_{1} = z_{3} - z_{4}

and

z_{4} - z_{1} = z_{3} - z_{2}

: each pair of opposite sides has the same displacement vector. Thus the given quadrilateral is a parallelogram. The identities also describe degenerate configurations if coincident or collinear vertices are allowed.

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