At the question Is there any equation for triangle? (MSE) the answer given by Henning Makholm received the most upvotes.
Therefore let's define the following triangle function $H$ with (a,b,c) the vertices of the triangle and $x$ an arbitrary point in the plane (all vectors): $$ H(x) = \big(|x-a|+|x-b|-|a-b|\big) \big(|x-b|+|x-c|-|b-c|\big) \big(|x-c|+|x-a|-|c-a|\big) $$ Due to the triangle equality, the outcomes between the parentheses $\big(\big)$ are always positive or zero, hence the product $H(x)$ is always positive or zero, regardless of the following question: is $x$ inside or outside the triangle ? The triangle function $H(x)$ only discriminates between $H(x)=0$ at an edge or $H(x)>0$ not at an edge of the triangle.

Now take a look at a common equation describing a circle with midpoint (vector) $a$ and radius $R$. Let $x$ be a point (vector) in the plane. Define the circle function C(x) as: $$ C(x) = |x-a| - R $$ Then for points at the circle $C(x) = 0$, for points inside the circle $C(x) < 0$ and for points outside the circle $C(x) > 0$. The circle equation divides the plane.
Note. It is of course true that dividing the plane can be circumvented by proposing an equation like: $$ C(x) = (|x-a| - R)^2 $$ But it seems not to be quite natural to do so.
Next take a look at the common function equation $y = f(x)$ , which can also be written as: $$ z = F(x,y) = y - f(x) = 0 $$ The (graph of the) function $y = f(x)$ divides the plane in a positive and a negative region.
And such is also the case with many common equations for straight lines, parabolas, hyperbolas, ellipses and so on and so forth. Still another example is in Fermat's proof for $x^3 - y^2 = 2$ : $$ f(x,y) = x^3 - y^2 - 2 $$ Then for example $(x,y) = (0,0) \rightarrow f(x,y) < 0$ and $(x,y) = (2,2) \rightarrow f(x,y) > 0$. The equation divides the plane.

But Henning's triangle equation does not divide the plane. Another triangle equation has been proposed as an answer to the abovementioned question and this equation does divide the plane.

Question. Should a "good" equation $F(x,y) = 0$ divide the plane ? Or is it just my personal bias ?

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    $\begingroup$ It is clear that what is meant by "good" equation is one which gives a properly embedded dimension $1$ submanifold. By a version of the Jordan-Brouwer separation theorem, then, the answer is yes. $\endgroup$ – Dustan Levenstein Nov 19 '13 at 15:50
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    $\begingroup$ More seriously, though, does $F(x,y) = x^2+y^2$ count as a "good" equation? $\endgroup$ – Dustan Levenstein Nov 19 '13 at 15:51
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    $\begingroup$ To formalize "divide the plane": We have continuous $F\colon\mathbb R^2\to\mathbb R$ and $F^{-1}(0)=\partial F^{-1}((-\infty,0)) = \partial F^{-1}((0,\infty))$. Yes, having such an equation is nice because it is numerically less affected by rounding errors and it quickly allows to determine interior vs. exterior points (if $F$ describes a connected simple curve) $\endgroup$ – Hagen von Eitzen Nov 19 '13 at 16:21
  • $\begingroup$ @DustanLevenstein: $F(x,y) = 0$ is a circle with radius $0$ and therefore a rather complicated way of defining the origin $(0,0)$. But OK, there's nothing "wrong" with it as such. $\endgroup$ – Han de Bruijn Nov 19 '13 at 18:57
  • $\begingroup$ @HagenvonEitzen: How about e.g. (connected) self intersecting curves? $\endgroup$ – Han de Bruijn Nov 19 '13 at 18:59

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