>The explanation of stencil or ancestor still begs the
>question: How do you know it is a circle?

Exactly same way that you know that a straight line can be darwn with a straightedge.

Keep in mind that
>Geometry is not about the (mechanical) drawing; but rather,
>it is about the mathematically exact line(s) or circle(s) it
>represents (theory).

Absolutely, I am sure you have imagined a straightedge. Now try imagining a curcular stencil.

>We know that when we draw a circle or
>arc with a compass that we are representing a theoretically
>exact circle because we are representing every point on the
>circle edge is being equidistant from the circle center
>point.

Yes, this is a common convention.

>Perhaps a better way to present the problem is "Is it
>possible to find the circle center point without using the
>circle center point and only using a straightedge."

Whatever does it for you.

>My whole issue with this problem and its proof is that with
>only an existing circle and straightedge one can only draw
>random lines.

This a sweeping claim. One may try drawing the lines purposefully, with a certain idea/goal in mind.

>> Why, it did a lot of sense to several generations of geometers
>> starting in the mid 1800s.
>
>Amateur geometers or Professional (w/ degree) Geometers?

Probably amateurs too but I know only about the professional geometers because of the literature they left.

>This might have been of great
>interest to amateurs, but I doubt this was of any interest
>to professionals.

If you look into 100 Great Problems of Elementary Mathematics by H. Dorrie, you'll see that #34 is call "Steiner's Straightedge Constructions". The theorem at hand is also Steiner's, I believe.

>> It certainly takes imagination rather than brute force.
>
>Imagination without reason does nothing to .

What about the brute force?