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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!

Laerion
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  • If you want a geometry software, try Geogebra. – user21820 Jun 07 '16 at 02:53
  • @user21820 Geogebra looks great! I fixed my question, hope is more clear now. – Laerion Jun 07 '16 at 03:02
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    If you want to get a feeling for Euclid's axioms, it would probably be best use Euclid's tools: A straightedge, and a compass! Actual, physical things that have been used for thousands of years. – pjs36 Jun 07 '16 at 04:40
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    @pjs .And be careful not to do what countless students have also done for thousands of years : Stab your fingertip with the pointy end of the protractor (drawing compass). :) – DanielWainfleet Jun 07 '16 at 05:20

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Expanding on pjs36's comment about using a straightedge and a compass. I cannot highly enough recommend the IOS app "Euclidea" and after that "Sketches" by the same developer. It's a joy to solve and find optimal solutions for the same sort of constructability (with only a straight edge and a compass) problems that Euclid may well have struggled with. It also builds reasonably steeply to being fiendishly hard and full of variety from the simplest premises. The premise always being to simply construct this…

alan2here
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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.

user21820
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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.

fleablood
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    Actually the equation for a circle (or ellipse) is a polynomial equation, at least according to standard definitions. – user21820 Jun 07 '16 at 04:01
  • And your first postulate is flawed because you need to handle the case when $x_1 = x_2$. Also, Euclid's idea of "line" is more like "finite line segment" rather than the modern notion of the infinite line. – user21820 Jun 07 '16 at 04:02
  • Actually my equation with the points as a factor of t take x1=x2 into account. I included the second only because it's more familiar. I'm well aware Euclid considered a line to be finite. That's why a separate equation for postulate 2 wasn't nescessary. But if you like $t\in [0,1] $ for 1. $t\in \mathbb R $ for 2. – fleablood Jun 07 '16 at 05:06
  • You're right that I didn't consider that the circle equation is a polynomial equation. But if we're going to quibble I will point out the op didn't ask for perinomial equations. Equations (polynomials) was asked for. – fleablood Jun 07 '16 at 05:12
  • I don't know what "perinomial" is supposed to mean. In standard algebra $x^2+y^2 = r^2$ is a quadratic polynomial equation in $x,y$. You can read about https://en.wikipedia.org/wiki/Polynomial_Diophantine_equation (and https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem for fun). – user21820 Jun 07 '16 at 06:45