I've a problem with Euclid's axioms. I understand them, but now I want some equations (polynomials) that I can use to draw some graphics and probe these axioms.
For example, a rect equation that probes Euclid's third axioms: "To describe a circle with any centre and distance [radius]."
Something that I can input in an app and get a graph of that, is that even possible? I mean, is there some app that can draw that or I've to do it by hand.
Thanks a lot!
On
Postulate (not axiom)1: two points $(x_1,y_1), (x_2,y_2) $, can determine a line. Line:${(t (x_2-x_1)+x_1,t (y_2-y_1)+y_1)}$ or $y=x\frac{y_2-y_1}{x_2-x_1}+(y_1-x_1\frac {y_2-y_1}{x_2-x_1}) $ if you prefer.
Postulate 2: to extend a line. Same as 1.
Postulate 3: to describe a circle by center $(a,b) $ and radius $r $. $(x-a)^2 +(y-b)^2=r^2$ (not a polynomial).
Postulate 4: all right angles are equal. Line 1: y=mx +b a line perpendicular will be be $y=-\frac 1m x +c $ for some c. (All perpendicular lines have same slope. And products of slopes of perpendicular lines are always -1.)
Postulate 5: line 1: y=mx + b and line 2: y=nx +c. If $m\ne n $ then line and line two have one point $(\frac {c-b}{m-n},\frac {c-b}{m-n}m+b) $ in common.
You can do similar for common notions.
Your question is (still) not very clear, because you're asking how to probe the axioms, but almost all geometry software (such as Geogebra) is based on assuming those axioms. For example, it might allow you to select any two points and draw a circle centred at one passing through the other. That can be partly justified by the fact that we can describe the circle by a Cartesian equation, but then to prove that all the points satisfying the equation are at equal distance from the centre we need some assumptions such as Pythagoras' theorem.