Construction Site
Using just a straightedge and a compass, the ancient Greeks devised methods for building basic shapes. Today computer programs like The Geometer's Sketchpad and Cabri Geometry create and animate complex shapes, with nary a ruler in sight



Straightedge and Compass The Elements, Euclid's classic work on geometry, begins with the construction of an equilateral triangle, shown at right. First, draw a line segment AB, then construct two circles with centers at points A and B, using AB as the radius. Next, draw lines from A and B to point C, one of two points at which the circles intersect. ABC must be an equilateral triangle because all three sides are radii of same-size circles. If you draw a line from C to D, you can find the midpoint of segment AB; CD is also perpendicular to AB.

Construct the figures below using just a straightedge and a compass, in as few steps as possible. (The equilateral triangle, for instance, uses three lines and two circles, for a total of five steps.) You may not eyeball measurements; you can draw a line or circle through any intersection of a line or circle.






Dynamic Geometry The computer program The Geometer's Sketchpad lets users trace how one point in a construction moves when another point is moved. How well can you follow a moving point? Can you figure out what path the red and purple dots will trace in the constructions below as the blue dot travels along the dotted blue line? Assume that wheels roll without slipping and that bars pivot without colliding or impeding one another's motion. The gray blocks are stationary.




Glad to Halve You In 1995 high school geometry teacher Charlie Dietrich of the Greens Farms Academy in Connecticut gave his students David Goldenheim and Dan Litchfield the routine challenge of dividing a line segment into any given number of equal pieces. Using The Geometer's Sketchpad, they responded with a construction he had never seen. Their discovery, which they named the GLaD construction, made headlines. Although they later found out that the basic idea was already known, their work demonstrates how new tools can spark new approaches. The steps of the construction are outlined below.


1.   To divide a segment AB into thirds, construct a rectangle on top of the segment and draw two diagonal lines, one connecting a bottom corner to the midpoint M of the top side of the rectangle and the other running from the other bottom corner to the opposite corner. Mark the intersection of the two diagonals P and drop a vertical line that intersects segment AB at point Q. Point Q divides AB into 1/3 and 2/3 lengths. Can you explain why? Hint: Triangles with matching angles have proportional side lengths.

2.   Here is the next step in the GLaD construction. Into what fractions does point R divide segment AB?

3.   Now continue with the same method. Into what fractions do points S and T divide segment AB?

4.   Here's a variation. Instead of dropping the first perpendicular from P, drop it from the midpoint M. Into what fractions do points Q, R, S, and T divide segment AB?



Solution

Want to see the solution to this puzzle?

Got new solutions for the puzzle? Want to see other people's solutions? Talk to the puzzle master in his discussion forum at www.scottkim.com.



For more puzzles and inventions by Scott Kim, including an archive of his past Bogglers, see his Web site, www.scottkim.com, or his book Inversions, Key Curriculum Press, 1996.

For more about Cabri Geometry, see www. cabri.imag.fr; for more about The Geometer's Sketchpad, see www.keypress.com/sketchpad.



© Copyright 2003 The Walt Disney