Dividing Line Segments in Affine Geometry

Here are the steps to divide a line segment into nn equal segments using only the tools in affine geometry. Here, we assume the axiom that there is a unique line through a given point that is parallel to a given line, i.e. the two lines do not meet.

  1. Construct the line segment AB\overline{AB}, given points AA and BB in affine nn-space.

  2. Take one of the points on the line segment, say AA, and draw a line through AA not parallel to the line ABAB.

  3. Pick a point C1C_1 on the line constructed in Step 2.

  4. Draw a line through AA not parallel to the line ABAB or AC1AC_1, and pick a point D1D_1 on this line.

  5. Construct a parallel line to the line AC1AC_1 through the point D1D_1.

  6. Construct a parallel line to the line AD1AD_1 through the point C1C_1, and define the meet (intersection) of this line with the line in Step 4 to be the point D2D_2.

  7. Draw the line D1C1D_1C_1 and construct a parallel line to this line through the point D2D_2.

  8. Define C2C_2 to be the meet of the line constructed in Step 6 with the line AC1AC_1.

  9. Repeat Steps 5-7, now with C1C_1, C2C_2 and D2D_2 instead of AA, C1C_1 and D1D_1; we do this n1n-1 times, so that we get nn copies of the line segment AC1\overline{AC_1}.

  10. Suppose then that the last point we have is CnC_n; we then draw the line BCnBC_n.

  11. Through each of the points CiC_i, for ii ranging from 11 to n1n-1, draw the lines through each of the points, parallel to the line drawn in Step 9.

  12. The meets of each of the lines in Step 10 and the line segment AB\overline{AB} will then be denoted by EiE_i, for ii ranging from 11 to n1n-1.

As a result of this construction, we then have that the points EiE_i, for ii ranging from 11 to n1n-1, divide the line segment AB\overline{AB} into nn equal line segments.

I did a construction (see picture above) where I just divided a simple line segment into 3 3 equal segments.

EDIT: Picture too small to look at by eye; not sure how to make it bigger. Just click to enlarge, I guess.

EDIT 2: There are a lot more things to say about this, but I am going to write a book with this content in it. Don't want to spoil too much...

Note by A Former Brilliant Member
2 years, 11 months ago

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