Specialist Mathematics · Unit 2 · Circle and geometric proofs · Geometric proofs using vectors
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Let OABC be a parallelogram with vertices at O, A, B, and C. Let vector OA = **a** and vector OC = **c**. The parallelogram has |**a**| = 5, |**c**| = 3, and **a** · **c** = 6. Prove that the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of all four sides. That is, prove that |OB|² + |AC|² = 2(|**a**|² + |**c**|²), and verify this result using the given values.
Prove that for any parallelogram with adjacent sides represented by vectors \(\vec{a}\) and \(\vec{b}\), the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides.
Prove that for any parallelogram \(ABCD\), the sum of the squares of the lengths of the diagonals equals the sum of the squares of the lengths of the sides. That is, prove \(|\overrightarrow{AC}|^2 + |\overrightarrow{BD}|^2 = 2(|\overrightarrow{AB}|^2 + |\overrightarrow{AD}|^2)\).