How does Pascal’s theorem apply to any conic section?

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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.

Pascal’s theorem states that if six distinct points are chosen on a circle and connected in a specific order to form three pairs of lines, the three points where these lines intersect are collinear.

In projective geometry, this theorem applies to any conic section because all conic sections—including the ellipse, parabola, and hyperbola—can be viewed as projective images of a circle. To demonstrate this for any conic:

Select six points on the conic section.
Project these points back onto a base circle.
Because the theorem holds for the circle, the three resulting intersection points are collinear.
Project these nine points (the six original points and the three intersection points) back to the conic section.

Since collinearity is an invariant property under projective mappings, the three intersection points remain collinear on the conic section. This approach effectively unites the different types of conics under a single geometric principle.