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Exercises for Section 2.2 Fano's Geometry and Young's Geometry Exercises [6] - [12] are about Fano's Geometry, introduced in Section 2.2.1 on page 36. [6] Prove Fano's Geometry Theorem #1. (presented in Section 2.2.1, on page 36.)

Answer :

Fano's Geometry Theorem #1 states: In Fano's Geometry, for any two distinct points A and B, there exists a unique line containing both points.

To prove this theorem, we need to show two things: existence and uniqueness.

Existence:

Let A and B be two distinct points in Fano's Geometry. We can construct a line by connecting these two points. Since Fano's Geometry satisfies the axioms of incidence, a line can always be drawn through two distinct points. Hence, there exists at least one line containing both points A and B.

Uniqueness:

Suppose there are two lines, l1 and l2, containing the points A and B. We need to show that l1 and l2 are the same line.

Since Fano's Geometry satisfies the axiom of uniqueness of lines, two distinct lines can intersect at most at one point. Assume that l1 and l2 are distinct lines and they intersect at a point C.

Now, consider the line l3 passing through points A and C. Since A and C are on both l1 and l3, and Fano's Geometry satisfies the axiom of uniqueness of lines, l1 and l3 must be the same line. Similarly, the line l4 passing through points B and C must be the same line as l2.

Therefore, l1 = l3 and l2 = l4, which implies that l1 and l2 are the same line passing through points A and B.

Hence, we have shown both existence and uniqueness. For any two distinct points A and B in Fano's Geometry, there exists a unique line containing both points. This completes the proof of Fano's Geometry Theorem #1.

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