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anniecvc
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Homework Statement
a) Suppose that A,B,C,D are four "points" in a projective plane, no three of which are on a "line." Consider the "lines" AB, BC, CD, DA. Show that if AB and BC have a common point E, then E = B.
b) From a) deduce that the three lines AB, BC, CD have no common point , and the same is true of any three of the lines AB, BC, CD, DA.
Homework Equations
Axioms of a projective geometry:
1) Any two "points" are contained in a unique "line"
2) Any two "lines" contain a unique "point"
3) There are four different "points," no three of which are in a "line"
The Attempt at a Solution
I proceeded by contradiction. Assume E is not equal to B.
AB and BC have common point E by assumption so ABE are on a line and BCE are on a line. (I was tempted to say aha! contradiction - 3 points on a line right here, but it's not illegal, only illegal for the 4 points A,B,C,D to have 3 on line.)
The E must connect to D by a line via Axoim 1.
Either ED is along line DCE or DAE.
If along DCE => B,C,D are along the same line
if along DAE => D,A,B are along the same line.
This contradictions that no 3 of the 4 points A,B,C,D are on a line.
Therefore E must be equal to D.
Don't think my argument is sensical.
And, for part b, I'm confused since doesn't Axiom 2 state that any 2 lines must contain a point?