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Section 1.2 Completeness
Definition 1.2.1. Fano Geometry.
Use the following axioms and definitions of intersection and parallel as a definition of the Fano geometry.
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There exists at least one line.
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There are exactly three points on every line.
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Not all points are on the same line.
Checkpoint 1.2.2.
Explore the Fano geometry as follows.
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Draw a line using Geogebra.
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Add a third point to the line.
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Note that Axiom 3 requires one more point. Draw one.
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Must any more lines be added? If so, do so.
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How many lines are in this geometry?
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Add the axiom: every point is on at least one line.
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To your line with three points, and one point not on that line add any lines required by this new axiom.
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Make sure these lines satisfy Axiom 2.
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How many lines are in this geometry?
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May any more lines be added? If so, do so, and be sure the axioms are satisfied.
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How many lines are in this geometry?
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What is needed to fix the difficulty noted in the previous question?
