I'm trying to figure out how to trisect a segment using only pen and ruler. There is a parallel line provided.

No measurement is allowed.

  • $\begingroup$ I think you need a compass too..!! $\endgroup$ – Vidyanshu Mishra Oct 9 '17 at 20:40
  • $\begingroup$ If we are given one fixed circle with its center included, we can do everything with just a straightedge. en.wikipedia.org/wiki/Poncelet%E2%80%93Steiner_theorem $\endgroup$ – Will Jagy Oct 9 '17 at 20:40
  • $\begingroup$ Do you have a given line parallel to the segment? $\endgroup$ – Oscar Lanzi Oct 9 '17 at 20:45
  • $\begingroup$ @OscarLanzi yes i do i forgot about it. $\endgroup$ – Kazz Oct 9 '17 at 20:49
  • $\begingroup$ What is a "metric oc."? Please describe the problem fully; there's no need to be terse here. $\endgroup$ – Chris Culter Oct 9 '17 at 21:02

lets mark ends of the segment as A and B.

  • mark points C and D on the parallel line.
  • draw lines BD and AC, mark the cross point as E
  • draw lines BC and AD, mark the cross point as F
  • draw line EF, mark its cross point with line CD as G
  • draw line BG, mark its cross point with line AD as H

The line EH cuts segment AB in 2:1 ratio.

  • $\begingroup$ can this be done easier (faster) ? $\endgroup$ – Kazz Oct 9 '17 at 21:13
  • $\begingroup$ Beat me to it, +1. $\endgroup$ – Oscar Lanzi Oct 9 '17 at 21:24
  • $\begingroup$ I know of no faster method without compasses. $\endgroup$ – Oscar Lanzi Oct 9 '17 at 21:27
  • $\begingroup$ What is the parallel line? Is it a line that happens to be parallel to AB some distance away? $\endgroup$ – corsiKa Sep 20 '19 at 18:14
  • $\begingroup$ yes, just as the name suggest :) $\endgroup$ – Kazz Sep 23 '19 at 5:26

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