Proposition 2.4 Proof

[Brandon Batchelor]

Proposition 2.4: For every point there is at least one line not
passing through it.

Proof:

Given the point P,and the distinct non-collinear points A,B,C.
By Prop. 2.2, <->AB,<->AC, and <->BC are concurrent.

Suppose P is equal to either of the points A,B,C, say A. Then <->BC
does not pass through P.

Suppose P is distinct from either A,B, or C. Then by I-3 at most two
of the lines <->AB,<->AC, and <->BC pass through the point P.

[#]

I am a little unsure if my use of prop. 2.2 is correct.

[Brandon Batchelor]

*Revision*

Proof:
Given the point P,and the distinct non-collinear points A,B,C.
By Prop. 2.2, <->AB,<->AC, and <->BC are concurrent.

Suppose P is equal to either of the points A,B,C, say A. Then <->BC

does not pass through P. The other two cases are easily verified.

Suppose P is distinct from A,B, and C. Then by I-3 at most two

of the lines <->AB,<->AC, and <->BC pass through the point P.
[#]

I am still unsure about this proof. Mainly the third part and the use
of 2.2.