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What does Pappus’s projective theorem say about six points on two lines?

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Projective drawingThe sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady. — projective geometry Projective drawing
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.

In projective geometry, Pappus’s theorem describes a fundamental relationship between points on two distinct lines.

Specifically, if you have two distinct lines and choose three distinct points on each—let these points be $A, B, C$ on the first line and $D, E, F$ on the second—you can define three intersection points based on the cross-connections of these points:
* Point $x$ is the intersection of lines $A E$ and $B D$ .
* Point $y$ is the intersection of lines $A F$ and $C D$ .
* Point $z$ is the intersection of lines $B F$ and $C E$ .

Pappus’s theorem states that these three resulting intersection points ( $x, y,$ and $z$ ) are always collinear, meaning they lie on a single straight line.1