I need to prove the following theorem geometrically and I have no idea how to even conceptualize what is happening.

"OX and OY are two straight lines and along OX five points 1,2,3,4,5 are marked at equal distances. Through these five points, parallels are drawn in any direction to meet OY. Prove geometrically that the third parallel is the mean of all five."

Any help on this would truly be appreciated!


enter image description here

Prove that then blue segment's length is the average of the lengths of all $5$ of the segments, assuming that the green lines are parallel.


  • $\begingroup$ Congratulations on seeing that interpretation. The description seemed rather vague to me. $\endgroup$ – Blue Sep 30 '17 at 22:56
  • $\begingroup$ Thank you! I couldn’t see it at all but now I do! $\endgroup$ – Michelle Drolet Oct 1 '17 at 0:35
  • $\begingroup$ By the way. This would be true if we didn't know the lines intersect at $O$. It'd be true if the lines were parallel and if ... well, I suppose the lines couldn't intersect in the middle of the 5 points (actually they could if we could consider segment on opposite sides of a line as having positive and negative values). $\endgroup$ – fleablood Oct 1 '17 at 1:19

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