Ring Theory as Applied to the Three Classical Geometric Construction Problems

The Rules of the Game
The goal is to geometrically construct specific shapes and lengths using nothing more than a straight edge and a compass.  The straight edge may only be used to draw a line between two points or to extend an already existing line, but it may not be used to measure lengths.  The compass may be thought of as a piece of string.  It's ends may be placed on two points and then a circle of radius equal to the distance between them may be drawn centered on either of the two starting  points.  The compass cannot directly mesure lengths either, but the fact that all points of a circle are equidistant from its center can be used to essentially "translate" a length.  If this is not clear now hopefully it will become so in the examples to follow.

So long as the shapes and points constructed are built using this framework any mathematics, no matter how modern, may be used to prove their properties.

Some Examples of Geometric Proofs

Below are links to several geometric constructions theorems.  Click on them to see a proof -- see if you can solve those that don't have proofs yourself.
  1. Given a line L and a point P on that line another line passing through P perpendicular to L can be constructed [Proof]
  2. Given a unit length (one) the integers can be constructed.[Proof]
  3. Given a point P and a line L another line can be constucted going through P and perpendicular to L.
  4. Given a point P and a line L another line can be constucted going through P and parallel to L.
  5. The rational numbers can be constructed.[Proof]
  6. Given any two numbers (lengths) A and B their product, (A x B), can be constructed.
  7. Given any two numbers (lengths) A and B their sum, (A + B), can be constructed. [Proof]
  8. For any given line segment the point bisecting that line segment can be constructed.
  9. Given a number, x, the square root of x can be constructed.[Proof]
  10. Given an arbitrary rectangle a square of the same area can be constructed.[Proof]
  11. Given an arbitrary triangle a rectangle of the same area can be constructed.[Proof]
  12. Given two squares a third square of area equal to the sum of the first two can be constructed.[Proof]
  13. Given any finite polygon a square of equal area can be constructed.[Proof]
  14. Given two arbitrary angles an angle equal to their sum can be constructed.
  15. Given a right triangle a lune of equal area can be constructed.
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