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Prove an identity about the opposite vertices of a parallelogram

by RoRi

Consider the quadrilateral OABC in the plane with A and C opposite vertices of the parallelogram. Prove that

    \[ A = \frac{1}{2} (C-A) = \frac{1}{2}B. \]

Proof. Since this is a parallelogram we have B = A+C. Therefore,

    \begin{align*} && B &= A+C \\ \implies && \frac{1}{2} B &= \frac{1}{2} (A+C) \\ \implies && \frac{1}{2} B &= A + \frac{1}{2} C - \frac{1}{2}A \\ \implies && \frac{1}{2} B &= A + \frac{1}{2} (C-A). \qquad \blacksquare \end{align*}

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