What are finite points in Geometry?
I was reading this and they mention "Suppose that $P$ and $U$ are finite points having normalized barycentric coordinates $(p,q,r)$ and $(u,v,w)$." I am wondering what they mean by finite here, as finite is usually used to denote sets and not points.
The term “finite points” occurs in the context of projective geometry. There on distinguishes between finite points and points at infinity.
A $d$-dimensional projective space can be decomposed into a finite $d$-dimensional affine space and a $d-1$-dimensional projective space “at infinity”. Depending on context, the choice as to what you consider infinity may be more or less arbitrary.
In your context, you have barycentric coordinates, leading to a very specific choice of infinity. The canonical (affine) representation of a point in the plane using barycentric coordinates $(a,b,c)$ is $P=aA+bB+cC$ with $a+b+c=1$. I guess that's what the term “normalized” in your text is referring to. Using homogeneous coordinates, one can often ignore the constraint $a+b+c=1$, using scalar multiples of those vectors instead. So $(2a,2b,2c)$ denote the same point as $(a,b,c)$. Points at infinity are exactly those for which no scalar multiple satisfies $a+b+c=1$, i.e. points where $a+b+c=0$. All points with that property taken together form the line at infinity.
For comparison: There also are other conventions being used when the homogeneous coordinates are not interpreted as barycentric (or trilinear) coordinates. One common convention is $z=1$. So a point $(x,y)$ would get homogenized to $(x,y,1)$. And an arbitrary point $(x,y,z)$ in the plane would dehomogenize to $(x/z,y/z)$ if $z\neq0$, but would represent a point at infinity if $z=0$. So in this case points satisfying $z=0$ form the line at infinity.
So sometimes it is important to explicitely state what convention you follow, in order to avoid confusion. Referring to your coordinates as barycentric already does that.