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The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three points.

I know why there are three possibilities for the location of the fourth vertex (3 possible diagonals), but I do not know how to use this in order to find the possible fourth vertices of the parallelogram.

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  • $\begingroup$ Do the problem in two dimensions first. Specifically, pick 3 points in the $xy$-plane, and try to find the possible fourth points (the pictures are much easier here!). $\endgroup$ – pjs36 Jun 19 '16 at 6:08
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Hint: Let $AC$ be the desired diagonal with $B$ as the third point opposite of the desired fourth point $D$. Then since opposite sides of a parallelogram have equal lengths and are parallel, observe that: $$ D = A + \overrightarrow{AD} = A + \overrightarrow{BC} $$

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