Given a triangle $ABC$ with side lengths $AB=c, AC=b, BC=a$. After some geometric constructions to create a point $X$, can we always calculate the barycentric coordinates of $X$ as a triplet of function $f,g,h$ with variables $a,b,c$? That is, $X=(f(a,b,c): g(a,b,c): h(a,b,c))$ (here I use the notation $(x:y:z)$ for non-normalized barycentric coordinate).
For example. Construct triangle $A'B'C'$ such that $A, B, C$ are the midpoint of $B'C', C'A', A'B'$, respectively. Let $M_a, M_b, M_c$ be the midpoint arc of arc $BC, CA, AB$ not containing $A, B, C$. Let $X, Y, Z$ be the midpoint of $A'M_a, B'M_b, C'M_c$, respectively. After a bunch of construction, I can calculate the barycentric of point $X$ as
$$X\left( -(b+c)^2:(b+c)(2b+c)-a^2: (b+c)(2c+b) -a^2 \right)$$
I feel like there might be some kind of geometric construction that we cannot express the barycentric coordinates of the constructed point as functions of $a,b,c$.
Furthermore, in the above example, I realized that the coordinates of $X, Y$, and $Z$ are somehow permutated to each other.
My question is:
- Can we always express the coordinates of a point $P$ that is somehow constructed as a triplet of functions of variables $a,b,c$?
- Is there a formal definition for geometric constructions, and cyclic geometric construction (as in the example)?
- Can we have formal proof that the points under cyclic geometric constructions have related components in their coordinates?
Thank you for you reading.
Update 1: Uptill now, I somewhat understand how geometric construction with straight-edge and compass are defined in algebraic meaning. The field of the constructible points is a quadratic closure of $\mathbb{Q}$. If so, given three sidelengths of triangle $a,b,c$ that $a,b,c \in \mathbb{Q}$, the point $X$ we constructed belongs to the quadratic closure of $\mathbb{Q}$. Hence the point $X$ has the coordinates that can be calculated as functions of $a,b,c$. But this popped me another question: Does setting $a,b,c$ to be rational numbers make the original problem loss its generality as the original side lengths are real numbers?