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flaw with Euclid's first proposition

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Where is the flaw and how can it be fixed? If R^2 is taken as a model for the Euclidean plane, does this help resolve the difficulty? Explain.

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Euclid's first proposition is analyzed.

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Hello,

Please see the attached file.
What is the flaw with Euclid's first proposition
and how can it be fixed? If R^2 is taken as a model for the Euclidean plane, does this help resolve the difficulty? Explain

Euclid's first proposition is:
Proposition 1.
On a given finite straight line to construct an equilateral triangle.

Proof
Let AB be the given finite straight line.
It is required to construct an equilateral triangle on the straight line AB.

Describe the circle BCD with center A and radius AB. Again describe the circle ACE with center B and ...

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