How does Desargues’s theorem relate two triangles viewed from one point?

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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.

Desargues’s theorem establishes a fundamental relationship between two triangles, $A B C$ and $A^{'} B^{'} C^{'}$ , that are in perspective from a point.

When these triangles are situated such that the lines connecting their corresponding vertices— $A A^{'}$ , $B B^{'}$ , and $C C^{'}$ —all intersect at a single point, the theorem states that the points of intersection of their corresponding sides are collinear; that is, they all lie on a single line.1

In the context of projective geometry, this theorem is significant because it describes a property based purely on the incidence of points and lines, independent of distance or angle measurements. While the theorem originally required a modification to account for parallel sides, the development of projective space—by adding points at infinity to Euclidean space—allowed the theorem to be formulated consistently, as parallel lines are considered to intersect at a common point at infinity.1