Newsgroups: rec.puzzles From: hoey@pooh.tec.army.mil (Dan Hoey) Date: 1997/02/26 Subject: Re: Segment bisection with compass alone. ma...@oban.CS.Berkeley.EDU (Gurmeet Singh Manku) asked: > How do we find the mid-point of a line segment using only a compass? Barry Wolk replied: : I can do it by using the compass seven times. Can this be lowered? Slightly, as in the following modification of your construction. : Let the given segment be AB. The points are located approximately by : the following diagram: : G : C E : : A M B F : : D : H : : Circle 1 has centre A and radius AB. : Circle 2 has centre B and radius BA. : Circle 1 meets circle 2 at points C and D. Circle 4 has centre C and radius CD. Circle 4 meets circle 2 at points D and F. : Circle 5 has centre F and radius FA. : Circle 5 meets circle 1 at points G and H. : Circle 6 has centre G and radius GA. : Circle 7 has centre H and radius HA. : Circle 6 meets circle 7 at points A and M. : Then M is the midpoint of AB. Proof omitted. Note that circle 3 and point E are absent from this construction. Is this six-circle construction optimal? My case analysis of the fourteen possible determinate four-circle constructions suggests so. None of those four-circle constructions includes a circle through M, so no fifth circle could produce an intersection at M. The only problem with this proof is that possible steps like "draw an arbitrary circle through point A intersecting circle C in two points" are not included in my analysis. I doubt such a step would help, but I have no proof. Nor am I sure that Euclid would look favorably on such an operation. Dan Hoey Hoey@AIC.NRL.Navy.Mil