Trisecting a Line Segment (with world record efficiency!)
The immediate goal of this module is to discover the
shortest way to trisect a line segment. The
bigger goal invites students to compare and contrast a variety of possible
geometric constructions. The module
begins with a trisection using four lines and two circles, then shows how to
construct one with three lines and two circles, then with two lines and three
circles, then finally with four circles. We
next consider constructions using only circles (Mohr-Mascheroni) and finally
using only lines (Poncelet-Steiner). Proofs
are given for each construction. The
Maple files showing the minimality of various constructions are meant for the
connoisseur and not your students.
Some ideas for using this module:
can use Geometer’s Sketchpad to recreate a construction or the proof.
your best students try to construct their own proofs for a construction.
and contrast constructions: Which
is most clever? most obvious?
prettiest? Which is most mind boggling?
goal was to use the least number of lines and circles.
Challenge your students to find the longest trisection!
is the length of the shortest construction to find 3 or 4 or 5 or any whole number times a
line segment? An interesting
and nontrivial pattern!
is the shortest construction to find 1/5 or 1/7 or 1/11 of a line segment?
students try the “carpenter’s method” in the References.
can trisect a line segment using origami (see Lang or the Hull reference).
can trisect a line segment using a marked ruler, e.g., begin with two points
that are five or seven cms apart, and let them use a ruler with centimeter
students work through the analytic geometry of the various proofs, e.g., the 2 circle/3 line, the 4
circle, and the Mohr-Mascheroni proofs.
Villanova Home Page
Opening Trisection Page
revised 25 Sept 2009