Alhazen's Billard Problem: Solving a Geometric Puzzle with Algebra

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SUMMARY

The Alhazen's Billiard Problem is a geometric puzzle that cannot be solved using compass and straightedge construction due to the necessity of cube root extraction, which is impossible with these tools. Participants in the discussion emphasized the importance of understanding the algebraic representation of equal-length chords in a circle to grasp the problem's complexity. The conversation highlighted the need for a rigorous mathematical explanation, referencing historical problems like the trisection of an angle and the doubling of the cube. Overall, the discussion serves as a platform for clarifying misconceptions and enhancing comprehension of the problem's underlying principles.

PREREQUISITES
  • Understanding of Alhazen's Billiard Problem
  • Knowledge of compass and straightedge construction limitations
  • Familiarity with cube root extraction in mathematics
  • Basic algebraic concepts related to circle geometry
NEXT STEPS
  • Research the algebraic representation of equal-length chords in a circle
  • Study the historical context of the trisection of an angle problem
  • Explore the implications of cube root extraction in classical geometry
  • Examine the mathematical rigor behind Alhazen's Billiard Problem solutions
USEFUL FOR

Mathematicians, geometry enthusiasts, and students seeking to deepen their understanding of classical geometric problems and their algebraic solutions.

FulhamFan3
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Alhazen Billiard Problem

I don't get why this problem is impossible with compass/straightedge construction.

I mean can't you draw a line bisecting the two points and where that line meets the circle is the point on the circle your looking for?

I'm probably understanding this problem wrong.
 
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FulhamFan3 said:
Alhazen Billiard Problem
I don't get why this problem is impossible with compass/straightedge construction.
I mean can't you draw a line bisecting the two points and where that line meets the circle is the point on the circle your looking for?
I'm probably understanding this problem wrong.

It says right there on the page.It would imply extraction of cube root.It's the same as other problems in th history of mathematics and especially the trisection of an angle and the famous Delic problem,the one with the doubling of the cube.
Try searching for this problem to the referenced bibliography (the 3 books mentioned there).I'm sure you'll e given a plausible mathematically rigurous explnation.

Daniel.
 
I know a cube root extraction is impossible with compass and straightedge. I'm arguing that it isn't necessary for this problem.
 
FulhamFan3 said:
I know a cube root extraction is impossible with compass and straightedge. I'm arguing that it isn't necessary for this problem.

On what grounds??Do you think the guy who posted this reason on 'wolfram' site was an imbecil??Or the guys who wrote the books he inspired from??Maybe so,but you'd better come up with something reliable instead of his bull****.

Daniel.
 
I mean can't you draw a line bisecting the two points and where that line meets the circle is the point on the circle your looking for?

Try it. (Make sure to set up an asymmetric problem so you don't get lucky!)
 
On what grounds?

Or maybe, just maybe, FulhamFan3 is trying to learn something? Please tone down your attitude.
 
I figured out what I was doing wrong that would make my solution invalid. The site has no diagram showing what they did to get that formula. I had no idea how they came up with the formula and I came here to see if someone could explain it. The solution seemed obvious so i didn't see what the deal was. Thanks for not explaning anything and being a dick dex.
 
Everything you need to know to get the formula, you learned in Algebra II! (really!)

Probably the easiest place to begin is to figure out how to express, algebraically, the notion that two chords of a circle are equal in length. You pick how to represent the lines algebraically.
 

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