Any recorded Euclidean construction is a collection of an arbitrary number of \(g\) statements. Any \(g\) statement is a collection of an arbitrary number of \(g\) objects and operators.
If, while reading a \(g\) statement, we come across an object for the first time during the course of this particular construction then the statement means:
take action
or
construct
the object in question.
If, on the other hand, we come across a previously encountered object then the statement is a reference to an existing object.
With the above rule in place let us investigate the exact meaning of the following sample \(g\) statement:
$$A \in Ln(B, C)$$Start deciphering the meaning of the above statement from its right-most object and move leftward:
- if a point \(C\) does not exist in the current construction then pick an arbitrary point on a plane and name it \(C\)
- if a point \(B\) does not exist in the current construction then pick a different from \(C\) arbitrary point on a plane and name it \(B\)
- if a straight line passing through the points \(B\) and \(C\) does not exist at the current state of this construction then construct it
- if a point \(A\) does not exist in the current construction then pick an arbitrary point different from \(B\) and \(C\) anywhere on the straight line passing through \(B\) and \(C\) and name that point \(A\)
However, if all the objects in the above statement do exist in the current state of this construction then it simply states a geometric fact which is a result of the previously taken steps:
because of so and so the point \(A\) is located on the straight line passing through the points \(B\) and \(C\)
Alternatively, if all the objects in the above statement exist and this statement is a part of a bigger statement then its meaning is:
because the point \(A\) is ... then the following must be true"
We observe that any compound statement can be decomposed into a collection of simple statements:
$$B$$ $$C$$ $$Ln(B, C)$$ $$A \in Ln(B, C)$$which are read verbatim as:
- pick an arbitrary point on a plane and name it \(B\)
- pick an arbitrary point on a plane other than \(B\) and name it \(C\)
- construct a straight line passing through \(B\) and \(C\)
- pick an arbitrary point on the straight line passing through \(B\) and \(C\) and name it \(A\)
\(\blacksquare\)