Chaos and Fractal

There is an impossible triangle.

a+a=2a, so you may say that it is possible. However, it is impossible.

a^2+b^2=a^2, so b^2=0. The height must be zero. It is impossible.

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a+b=c, and b must be a. As I mentioned, it is ridiculous. However, if a+a=c=2a were possible, you would need to put fractal triangles in it.

a*=b*=a/2. c*=c/2=a

These are fractal triangles.
a* would be a/2,a/4,a/8,・・・a/2^x.
c* would be a,a/2,a/4,・・・a/2^x-1.
2a*=(a/2^x)×2=a/2^x-1=c*=a*+a*.
a*=a/2,so a/2+a/2=a.
∴a+a=2a=c
Therefore, an impossible triangle is possible. When x is ∞, a*=a/2^∞ would be close to zero. If you saw the zero point, you could see the line which includes infinite triangles. It must be fractal.