Prove $X^2+Y^2-1$ is irreducible using geometrical tools. I'm trying to understand what is meant in this paragraph:
 of "Conics and Cubics. A Concrete Introduction to Algebraic Curves (by Robert Byx)": 

He wants to prove that the polynomial $X^2+Y^2-1$ is irreducible using geometrical  tools. I have the following doubts:


*

*What he means by Since every line has at least three points on it ?

*Why the fact that $X^2+Y^2-1=F(X,Y)\cdot G(X,Y)$, where F, G are lines implies that the circle $X^2+Y^2-1=0$ is made up of two lines?
I would really grateful if anyone could help me.

The "proof" is absolute nonsense:   
If you replace the word "circle" by "degenerate conic" (= union of two lines, non necessarily distinct) the "argument" still applies, which proves that it is false, since degenerate conics do exist!
And actually the circle is reducible over an algebraically closed field of characteristic $2$, and such a field is necessarily infinite, so that lines have infinitely many points.
This confirms that  the purported proof is no proof at all.  
The mistake is the statement "a line and a circle have at most two points in common": this is false if the circle contains a line and since what we want to prove is that the circle contains no line we may not assume it in the proof: the argument is circular (pun intended!)
