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Inscribed circle problem

 
 
 
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SLD

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OK, imagine a simple semicircle of radius 1 centered above the origin. I.e. y=sqr(1-x^2). No imagine all possible circles inscribed inside that semi circle such that the x axis and the larger semicircle are tangents to those circles.

Thus at x=1 and -1 the radius of that circle would shrink to 0 and at x=0 the circle would be its largest with r=1/2.

If you traced the center of all such circles from -1 to 1, is there a formula for that line?

ChatGPT kept saying it is 1/2*sqr(1-x^2). Which is just 1/2 of the circle essentially. But that is clearly wrong. Try it for 0.8 and you will see.
 
Maybe I did something wrong but I get:

y = 1/2 *(1-x^2),

which is a parabola, right? Same as what you have without the sqrt.

The way i did it was imagine a circle for which the center is x from the origin and y above the x-axis. Then the center of that circle is on a radius of the big semicircle, so that the circle is tangent to the semi-circle. Then the line from the origin to the center has sides of x and y with a hypotenuse of 1-y.

Then just use Pythagoras theorem to solve for y

Is that correct?
 
Shadowy Man said:
Maybe I did something wrong but I get:

y = 1/2 *(1-x^2),

which is a parabola, right? Same as what you have without the sqrt.

The way i did it was imagine a circle for which the center is x from the origin and y above the x-axis. Then the center of that circle is on a radius of the big semicircle, so that the circle is tangent to the semi-circle. Then the line from the origin to the center has sides of x and y with a hypotenuse of 1-y.

Then just use Pythagoras theorem to solve for y

Is that correct?
Click to expand...
That looks right by my graph of the function.
 
From multiple reporting AI is not what people think it is.

It makes things up, and gets facts wrong.

It can be a useful tool but you cna't use it with blind faith.

The key word is 'artificial' mimicking human reasoning. Humans are not infalible, even experts in a field.
 
Last edited:
steve_bank said:
From multiple reporting AI is not what people think it is.

It makes things up, and gets facts wrong.

It can be a useful tool but you cna't use it with blind faith.

The key word is 'artificial' mimicking human reasoning. Humans are not infalible, even experts in a field.
Click to expand...
Yeah, I’m really stunned at the failure here. I had high hopes for chat gpt and AI. But it’s not real AI. It’s just mimicking human conversation, while fucking up entirely. It seems good at explaining some basic math problems. I’ve asked it to do some complicated differential equations and sometimes it actually does it. I’ve asked it about series convergences and it seems to get the correct answers. This one it just flubbed.

I’ve also used it to explain to me some dumb questions I had about mathematics that I have studied. I failed to realize that there was a difference between multiple complex variables and quaternions.
 
I solved this problem with analytic geometry and Mathematica. The coordinates of the inscribed circle's center point are {cx,cy} and those of the intersection with the outer circle are {px,py}
  • Outer circle: px2 + py2 = 1
  • Inner circle: (px-cx)2 + (py-cy)2 = py2
  • Tangent condition: {-py, px} . eps2 . {-(py-cy), (px-cx)} = 0
where eps2 is the 2D antisymmetric symbol {{0,1},{-1,0}}.

The solution:
  • px = 2*cx/(1+cx2)
  • py = (1-cx2)/(1+cx2)
  • cy = (1/2)*(1-cx2)

The center of the inner circle is thus on a parabola.
 
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ChatGPT seems to me to be hopelessly glib, talking a lot but not saying very much.
 
SLD said:
I won’t say it’s entirely useless. It seems to solve some things well.
Click to expand...
Seeming to solve some things well, while confidently spouting utter bollocks once in a while, IS entirely useless.

If the answers it gives are sometimes right and sometimes wrong, but always sound good, then the only time you can trust (and therefore use) them is when you already know the answers - in which case, you didn't need them.

So what, exactly, is it good for?
 
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