I wonder if there have been attempts to write the constructions that are performed in Euclidean geometry – drawing straight lines and circles defined by two given points, using specific generated intersection points for the generation of further lines and circles – in an algebraic (symbolic) way, such that they can be (i) related to algebraic expressions for numbers (resp. lengths) or can be (ii) algebraically transformed into another expression that yields equivalent results.
The easy part is to symbolize the line through two (different) points $A$ and $B$ by a symbol like $-AB-$ which denotes the same as $-BA-$. The line segment from $A$ to $B$ may be symbolized by $AB\rangle$ which is different from $BA\rangle$. Let $|AB\rangle|$ be the length of $AB\rangle$. The circle with center $A$ going through $B$ may be symbolized by $AB)$ which denotes the same as $(BA$ but another circle than $BA)$.
But there are two severe problems:
There may be no intersection point for a pair of lines and/or circles.
There may be two intersection points for a pair of circles or of a circle and a line.
So, when you symbolize the intersection point of two lines $L_1$ and $L_2$ by $L_1 /\!\!| L_2$, it might be the case that $L_1 /\!\!| L_2$ denotes nothing (or null in programmer's language) which is bad for algebraic expressions.
On the other hand, for being specific you must distinguish the two intersection points of two circles, e.g. like this: $C_1 \bigcirc\!\!\!\!\bigcirc^{\cdot} C_2$ and $C_1 \bigcirc\!\!\!\!\bigcirc_{\cdot} C_2$. But which one is which? This problem might be solved by using the orientation of the circles, $C_1 \bigcirc\!\!\!\!\bigcirc^{\cdot} C_2$ being the first intersection point when going clock-wise, $C_1 \bigcirc\!\!\!\!\bigcirc_{\cdot} C_2$ being the second – roughly speaking.
But how to solve the first problem (the null problem)? One might say: An algebraic expression for a Euclidean construction is invalid when it contains an intersection point which is null. The question would immediately arise: How to check, if an expression is valid? (Validity presumably cannot be defined syntactically, but note that the "invalidity" of numerical algebraic expressions like $x/0$ or $\sqrt{-1}$ cannot be defined syntactically, neither.)
And what, if the construction assumes that two intersection points $C_1 \bigcirc\!\!\!\!\bigcirc^{\cdot} C_2$ and $C_1 \bigcirc\!\!\!\!\bigcirc_{\cdot} C_2$ are different (allowing to draw a unique straight line through them), when they in fact are the same, i.e. there is only one intersection point of $C_1$ and $C_2$? This would make the expression invalid, too.
In any case, here is an example for the construction of the midpoint $M$ between two points $A$ and $B$:
The algebraic expression of $M$ is
$$M = -AB- /\!\!| - AB) \bigcirc\!\!\!\!\bigcirc^{\cdot} (AB\ \ AB) \bigcirc\!\!\!\!\bigcirc_{\cdot} (AB -$$
or in a better readable way:
$L_1 = -AB-$
$C_1 = AB)$
$C_2 = (AB$
$P = C_1 \bigcirc\!\!\!\!\bigcirc_{\cdot} C_2$
$Q = C_1 \bigcirc\!\!\!\!\bigcirc^{\cdot} C_2$
$L_2 = - PQ -$
$M = L_1 /\!\!| L_2$.
Now continue and construct
$C_3 = (AM$
$N = L_1 \emptyset_\cdot C_3$
i.e.
$$N = -AB- \emptyset_\cdot (A -AB- /\!\!| - AB) \bigcirc\!\!\!\!\bigcirc^{\cdot} (AB\ \ AB) \bigcirc\!\!\!\!\bigcirc_{\cdot} (AB -$$
It turns out (is obvious) that $N = B$, which may be stated with lengths $|\cdot|$ as
$$|AM\rangle| = |MB\rangle| = \frac{1}{2} |AB\rangle|$$
which in turn would allow to symbolically abbreviate the expression for the construction of the midpoint $M$ of $AB\rangle$ by
$$\frac{1}{2} AB\rangle := -AB- /\!\!| - AB) \bigcirc\!\!\!\!\bigcirc^{\cdot} (AB\ \ AB) \bigcirc\!\!\!\!\bigcirc_{\cdot} (AB -$$
So my question is:
Has a way been found to circumvent the problems stated above, and is there a known algebraic notation of geometric constructions in the spirit of Euclid (which may be obsolete today)?