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Let there be a circle $(O,R)$ and $AB,CD$ two perpendicular chords of that circle that intersect on point $E$. Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$

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First prove it parallel to $\vec{AB}$, then prove it parallel to $\vec{CD}$.

To prove it parallel to $\vec{AB}$, say, let $X$ be the foot of the perpendicular from $O$ to $\vec{AB}$, then express $\vec{EA}$ in terms of $\vec{AX}$ and $\vec{EX}$, and $\vec{EB}$ in terms of $\vec{BX}$ and $\vec{EX}$; then use the fact that $\vec{AX} = \vec{BX}$.

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    @nick: We have a vector equation in the form $\vec{x} = \vec{y}$. By "prove parallel to $\vec{AB}$", I meant: prove that the component of $\vec{x}$ in direction $\vec{AB}$ is equal to the component of $\vec{y}$ in direction $\vec{AB}$.2011-09-20