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Earlier I wrote:
"More later today! I promise. :)"

Did I say that? I meant tomorrow, not today. But actually tomorrow is already today, so back on track for "today", I guess.

Here's a key piece of it... For "thin" triangles with short side or "base" of length b, and height equal to h, when the ratio h/b is considerably greater than one... effectively defining a "thin" triangle, the true least squares fix is located just above the mid-point of the base at a height, y, relative to the base length, b, equal to half the base divided by the height:

•  y / b = (1/2) b / h

This is an exact result for the limiting case where b/h is small and a surprisingly accurate result for much more moderate values of b/h. In words:
  "the true least-squares fix in a thin triangle of three lines of position is located just inside the triangle above mid-point of the short side with height above the short side relative to the length of the short side equal to one-half the short side length divided by the long side length"

Still not clear? Just draw a skinny triangle and think it through! It's easy. Genuinely easy. :)
This is a big deal. No joke. This clean, short, simple result is completely consistent with the computation rules that have been established for decades (e.g. in the back of the Nautical Almanac) and completely consistent with the technique of crossing "symmedians". This fast, simple analytic rule is specific to thin triangles, which are, by far, the most important cases. It's not hard to prove this rule for "thin" isosceles triangles by a little brute force --yet relatively simple-- calculus work. Similarly it is not hard to prove this for "thin" right triangles by a bit more calculus. But the real fun comes next... There is a really slick way of demonstrating this rule and a bunch of other fascinating transformations, in the specific case of "right" triangles for fixes. More on that tomorrow... today!

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA