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    Lines and Intersections

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    Incidence Geometry: Give proofs for the following:

    i.For every line there is at least one point not lying on it.

    ii.For every point, there is at least one line not passing through it

    iii.For every point P, there exist at least two distinct lines through P.

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    https://brainmass.com/math/geometry-and-topology/lines-intersections-226748

    Solution Preview

    Incidence Geometry
    Give proofs for the following:

    i. For every line there is at least one point not lying on it.
    Proof:
    We know that there exist three non-collinear points. Let us call these P,Q, and R. Let p be any line. Then at least one out of P, Q, R is does not lie l, since P,Q, and R all coincident with p contradicts the fact that they are non-collinear.

    ii. For every ...

    Solution Summary

    Step-by-step proofs are provided to all the lines and intersections statements.

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