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How to Divide a Line into Three Equal Parts Using an Unmarked Ruler and Compass (Four Methods):

Dividing a line into three equal parts can be achieved using an unmarked ruler and compass in four different ways. Let's break it down step by step—it's super simple!

Trisect a Line, Method 1:
1. Draw any line segment AB.
2. Use point A as the center and AB as the radius to draw an arc.
3. Use point B as the center and AB as the radius to draw another arc.
4. Identify the intersection points C and D.
5. Draw a line through points A and C, intersecting arc A to obtain point E.
6. Draw a line through points B and E.
7. Draw a line through points A and D.
8. Identify the intersection point F.
9. Draw a line through points C and F, intersecting AB to obtain point G.
10. Use G as the center, AG as the radius, and draw arcs intersecting GB to obtain point H.
11. Confirm that AG = GH = HB.

Trisect a Line, Method 2:
1. Draw any linesegment AB.
2. With A as thecenter, draw a circle with a radius smaller than AB.
3. Draw a diagonal linepassing through point A, intersecting at points C and D.
4. Draw a line passingthrough points C and B.
5. Use a compass tofind the perpendicular bisector of CB.
6. Locate the midpoint E of CB.
7. Draw a line passingthrough points D and E, intersecting line AB at point F.
8. With F as the centerand AF as the radius, draw points G where it intersects with FB.
9. Confirm that AF = FG= GB.

Trisect a Line, Method 3:
1.Draw any line segment AB.
2.With A and B as centers, draw two circles with radii equal to AB.
3.With the intersection of the two circles as the center, draw a circle with a radius equal to AB.
4.Mark the intersection points as C and D.
5.Draw a line passing through points B and C, intersecting circle C at point E.
6.Draw a line passing through points E and D, intersecting AB at point F.
7.Use a compass with F as the center and AF as the radius to draw a circle intersecting AB at point G.
8.Now, AF = FG = GB.

Trisect a Line, Method 4:
1. Draw any line segment AB.
2. With A and B as centers, draw arcs with radii equal to AB, intersecting at points C and D.
3. Draw a line through points B and D.
4. Draw a line through points A and C.
5. Use a compass to divide the line segments BD and AC into equal parts, obtaining points E, F, G, H such that AE = EF = GH = HB.
6. Draw lines through points H, F, and E, G.
7. On the line AB, mark points I and J.
8. Finally, you will have AI = IJ = JB.
9. Similarly, you can trisect the line segment at any desired points.

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