Maclaurin Construction of a Conic from 5 Points (projective).cdy

I suppose I should mention why I am interested enough in conics to post
two constructions of them. It is because these two constructions use
only projective geometry. And I only just discovered what conics are in
projective geometry, and I'm amazed.

After all, normally we learn of conics in terms of distances - an ellipse is
the locus of points whose distance is a constant sum of the distances from two
other points, a parabola is a locus of points the same distance from a line and
a point, etc. And the formulas we learn in analytic geometry for circles,
parabolas, etc. are based on distances.

But projective geometry has no concept of distance, so how can you have conics?

Well, the definition of a projective conic is: The locus of the points of
intersection of corresponding lines of two projective pencils is called a
point-conic. (And there is a corresponding definition for the dual,
a line-conic.)

So that is definitely a projective definition, but how could that possibly
mean the same thing as a definition based on distances?

Yet, this construction, which is based solely on that definition, generates
conics identical to the conics defined by distances, they are the same conics.
The next construction I posted makes conics based on Pascal's Theorem which
is also a theorem of projective geometry, no distances are used.

I find this to be amazing! So, I posted these constructions.

-- David