Line and Circle: The Straightedge and Compass Tools

Two of the geometry 'tools,' which do not depend on measurement, are the straightedge and compass. The straightedge allows us to join any two points with a line segment, and the compass allows us to construct a circle with any given point as center and with any given radius. You could think of these as construction tools, or as the realization of Euclid's Postulates #1 - 3.

These two tools are simple, but deceptively powerful in creating designs. Below is a design to try.

Daisy Design:

In the following construction, which is called a daisy design, lightly draw the circles you make in pencil so that you can erase parts of these circles to emphasize the design you want.

1. Mark a point A on a sheet of paper. Open your compass to any radius, and construct a circle with A as center. Mark any point B on the circle. Without changing the compass radius, construct a circle with B as center. The new circle will intersect the first circle in two points, say C and D. Construct circles centered at C and at D, each having the same radius as before. These circles intersect the original circle in new points, say E and F. Now construct circles centered at points E and F. Each of these circles intersect the original circle in a single new point, say G. As the final step in the design, construct a circle centered at point G, again with the same radius as before..

Portfolio Idea: Study the result of the constructions, and create from it some design of your own just by erasing arcs of circles and coloring sections of the circles. Do a study of designs created by using just a compass and coloring selected regions.

2. The fact that the constructed circles centered at points E and F intersect the original circle in a single new point makes the resulting design rather neat.

What would happen if you had changed the compass radius after constructing the first circle and you had constructed a circle with B as center but with this different radius? (And then continued as before wiith the second radius for all the subsequent circles)

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Explain why, under the initial directions with just one radius, the final two circles (the circles centered at points E and F) intersect in a single point, G, on the first circle. This problem is #13 on the Problems page.

Equilateral Triangle:

4. Imagine an equilateral triangle and think of relationships(s) that the three vertices have with each other.

Given two points, construct (using only straightedge and compass) a third point so that the three are vertices of an equilateral triangle. (Hint: Look at the daisy design.)

Show your construction process and state an argument to prove to someone that your construction "works."

Bisection of an Angle: (Bisection of an angle BAC means constructing a point F, so that angle BAF is congruent to angle FAC; see the diagram at Euclid's Proposition 9.)

5. For a given angle, use straightedge and compass to construct the ray that bisects the angle.

Show your construction process and state an argument to prove to someone that your construction "works."

Bisection of a Segment:

6. For a given line segment, use straightedge and compass to construct the point that bisects the segment.

Show your construction process and state an argument to prove to someone that your construction "works."

Perpendicular to a Line through a Point:

7. Imagine points called P and Q and the process of folding P onto Q; imagine a few of the points on the fold line that is constructed. Imagine also folding the line passing through P and Q.

Now, start with just any given line and any given point. Use straightedge and compass to construct the line perpendicular to the given line and passing through the given point.

Show your construction process and describe that process verbally.

Parallel to a Line through a Point:

8. Now, start with just any given line and any given point. Use straightedge and compass to construct the line parallel to the given line and passing through the given point.

Show your construction process and describe that process verbally.

Euclid and Measurement:

9. State at least one reason why one might want the results in a geometry to be independent of measurement.