Use the following information we know about Projective Geometry and the Extended
ID: 3109515 • Letter: U
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.