Adam's Circles: Splitting & Connecting Segments

  • #1
maxkor
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Adam has a circle of radius centered at the origin.

- First, he draws segments from the origin to the boundary of the circle, which splits the upper (positive ) semicircle into equal pieces.

- Next, starting from each point where a segment hit the circle, he draws an altitude to the -axis.

- Finally, starting from each point where an altitude hit the -axis, he draws a segment directly away from the bottommost point of the circle , stopping when he reaches the boundary of the circle.

What is the product of the lengths of all segments Adam drew?
 

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  • #2
Beer induced query follows.
maxkor said:
... What is the product of the lengths of all segments Adam drew?
Product or sum?
 
  • #3
jonah said:
Beer induced query follows.

Product or sum?
Product.
 
  • #4
Just to check whether I'm thinking along the right lines, should the answer be
?
 
  • #5
Beer induced reaction follows.
Opalg said:
Just to check whether I'm thinking along the right lines, should the answer be
?
I get the same; although mine's just an approximation, 0.000495504345414
Curious as to how you got an exact expression.
Did you use the math god Wolframalpha?
I was under the impression that while the endpoints of the green lines can be epressed exactly, I settled for an approximation. I guess I didn't took it far enough.
The product of the red and blue lines are of course
 
  • #6
maxkor said:
Adam has a circle of radius centered at the origin.

- First, he draws segments from the origin to the boundary of the circle, which splits the upper (positive ) semicircle into equal pieces.

- Next, starting from each point where a segment hit the circle, he draws an altitude to the -axis.

- Finally, starting from each point where an altitude hit the -axis, he draws a segment directly away from the bottommost point of the circle , stopping when he reaches the boundary of the circle.

What is the product of the lengths of all segments Adam drew?
The six points on the semicircle have coordinates . The red segments all have length and the th blue segment has length .

The th green segment lies on the line joining and , which has equation . That meets the semicircle when , which leads after a bit of simplification to the point So if is the length of the th green segment then Again after some simplification, this becomes , which I prefer to write as .

Putting together everything done so far, the product of the lengths of the 18 segments is To evaluate that product, notice that the numbers , together with , are the solutions of the equation . But (either by working with trig. identities or by using de Moivre's theorem) After discarding the solution , you see that are the solutions of . The product of the roots of that equation is . Therefore Next, putting you see that are the solutions of . That simplifies to , and the product of the roots is . Each value of corresponds to two (equal) values of , so we should square that answer. But then we want to take the square root (getting back to where we started from), for the formula Finally,
 
  • #7
It' correct.
 

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