Use the following information we know about Projective Geometry and the Extended

Question

Use the following information we know about Projective Geometry and the Extended Euclidean Plane (posted below) to help verify that the Extended Euclidean Plane is a projective plane by verifying axiom P3 for this model.

Info for Extended Euclidean Plane:

One A proyectve pline is an axiom systern uutn undufined terms "point line", and incidence Satis three (POi There are points, no mree them on a line (Ea). If P and Q are disinot points, nure is exacty are line, PO containing them both If t and m are distinct lines tmure Point P on bom Bemis by P3 are no parallel lines.

Explanation / Answer

Let l and m be two distinct, nonparallel lines.

By definition of nonparallel there exists at least one point, P, that lies on both l and m.

Suppose there exists a different point, Q, be on both l and m.

By P2, there exists exactly one line on which the two distinct points lie.

Since P and Q are on both l and m, then l and m are the same line.

But l and m are defined as being distinct lines which is a contradiction.

Thus, there exists exactly one point P such that P lies on both l and m.

hence P3 is verified.

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