How Many Ways to Trisect a Line?

Solution 1 (GSP Construction – Trisect Line 1)

1. Start with line segment AB

2. Construct a circle with center A passing through point B

3. Construct a circle with center B passing through point A

4. The circles intersect at two points, label these C and D

5. Construct a line segment/chord from one of the intersection points to each center of the circles/ends of line segment AB. For my construction I have chosen intersection point D.

6. Find the midpoints of the chords.

7. Connect the midpoints of the chords with the other intersection point (in this case point C)

8. The last construction trisects the line segment AB at points E and F

*The first four steps could be used to bisect any line segment if the intersection points of the circles were connected:

Solution 2 (GSP Construction – Trisect Line 2)

1. Start with line segment GH.

2. Create a ray from one end of the line segment, I have chosen ray GI

3. Create a circle with center at I passing through the point G; use the circle’s intersection of the ray, point J, to create another circle centered at J passing through I. Create a third circle, K, using the same radius.

4. Construct line segment KH by connecting the points, then construct parallel lines to line segment KH from each of the circle centers.

5. The parallel lines will trisect the line segment GH at L and M

*This construction could be used to divide a line segment into n equal parts.

Solution 3 – (GSP Construction – Trisect Line 3)

1. Start with line segment AB, then construct a circle with center A passing through point B; do the same for circle B.

2. Extend line segment AB to ray AB.

3. From the intersection point of the ray and circle B, create a new circle with the intersection point, C, as the center passing through point B.

4. Circle C will intersect the ray at another point, D.

This gives you a trisected line segment AD

*This construction gives you a way to create n equal parts multiple times:

I realize there are seven ways to trisect a line segment, like the problem asked, but this was all I could come up with!