Is a point at infinity unique?
When I was studying inversion, I thought so. Because inversion of center of circle is point at infinity. But I'm confusing after solving the following problem:
Problem: Let $ABCD$ be a quadrilateral. Define the points $P=AD\cap BC, Q=AB\cap CD$ and $R=AC\cap BD$. Let $X_1, X_2, Y_1, Y_2$ denote $PR\cap AD, PR\cap BC, QR\cap AB, QR\cap CD$. Prove that lines $X_1Y_1, X_2Y_2$ and $PQ$ are concurrent.
We choose projective transformation $ABCD$ sends to square $A'B'C'D'$. Then $P', Q'$ are point at infinity. If point at infinity is unique, we get $P'\equiv Q'\implies P\equiv Q$.
Name of the book: "Euclidean Geometry in Mathematical Olympiads" by Evan Chen.
Best Answer
There are differing answers depending on your level of study. The answer here will be in the context of what you need for the IMO.
Different geometries have different infinities. The basic idea is that if the operations/transformations allowed in a particular geometry lead to non-finite points, life will be simpler if you just add them. Although these infinite points may seem special, they can always be mapped into finite points using the transformations of the geometry.
In Euclidean geometry, the transformations are translations, rotations, and optionally dilations. There are no points at infinity. You don't need them.
In inversive geometry, the transformations are inversions in circles and lines. Inversion in a circle $c$ will be undefined for the center of that circle unless you have defined a point at infinity. A single point at infinity suffices. Note that you can transform this point to a finite point $p$ simply by inverting in a circle centered at $p$.
In projective geometry, the transformations are projective transformations. A general projective transformation $T$ will transform an entire line to infinity. So you need a line at infinity to accommodate all these points. This line at infinity (and its points) can be mapped back to a finite line $\ell$, which will intersect the line at infinity at a point at infinity that corresponds to the direction of the line. The comments to your question add a little more detail.
The bottom line is that the nature of infinity depends on the geometry you are working in. For IMO purposes, you have to understand that inversive geometry and projective geometry have different interpretations of "infinity".