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Prove that the diagonals of a rhombus are perpendicular

Prove that the diagonals of a rhombus are perpendicular.


Proof. For any rhombus, we can take the vertices to be the origin, and the points A,B, and A + B in \mathbb{R}^2. So, the diagonals are the vectors A+B and A-B. Then,

    \begin{align*}  (A+B) \cdot (A-B) &= A \cdot A - B \cdot B \\  &= \lVert A \rVert^2 - \lVert B \rVert^2 \\  &= 0 \end{align*}

since the edges of a rhombus are of equal length. \qquad \blacksquare

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