I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a "straight line"? If the curve is not a straight line then does it follow that the theorems that rely on such a construction will not necessarily be valid? BTW I am speaking of Euclidean geometry only.
[Math] a straight line
euclidean-geometrygeometry
Related Solutions
The solution below is based on the assumption that drawing a line parallel to the other requires no auxiliary circle(s).
Step 0: Select a random point A on the given circle.
Step 1: Draw chord AB of reasonable length.
step 2: Draw $C_1$ (centered at A, radius = AB).
Step 3: Locate C where BA produced cuts $C_1$.
step 4: The two circles will meet at D. CD, when joined, will cut the given circle at E.
Step 5: Note that $\angle CDB = \angle EDB = 90^0$ because BC is the diameter of $C_1$.
Step 6: Join BE which is then the diameter of the given circle.
Step 7: Through A, draw a line parallel to CD cutting BE at F. By intercept theorem, FB = FE.
[Note:- Drawing a line parallel to a given line can be done by translating "set squares, or straight rulers" without drawing auxiliary circles. Whether the construction of such line is considered as a 1-step construction or not depends heavily on the rule of the game.]
Best Answer
Euclid didn't define "line" by an axiom; in one translation, he defined it as "breadthless length," which is the kind of definition that only helps you if you already know what a line is. Isaac has it right when he suggests that what's important about a line is what it does, not what it is. An alternative approach is to found Euclidean geometry on analytic geometry. Define the real number line, then the Cartesian coordinate system, then define a line to be the set of solutions of $ax+by+c=0$ for fixed $a,b,c$ with $a$ and $b$ not both zero.