Geometry for Elementary School/Copying a line segment

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Geometry for Elementary School
Constructing equilateral triangle Copying a line segment Copying a triangle


This construction copies a line segment to a target point T. The construction is based on Book I, prop 2.

The constructionEdit

  1. Let A be one of the end points of . Note that we are just giving it a name here. (We could replace A with the other end point B).



  2. Draw a line segment



  3. Construct an equilateral triangle (a triangle that has as one of its sides).



  4. Draw the circle , whose center is A and radius is .



  5. Draw a line segment starting from D going through A until it intersects and let the intersection point be E . Get segments and .



  6. Draw the circle , whose center is D and radius is .



  7. Draw a line segment starting from D going through T until it intersects and let the intersection point be F. Get segments and .

ClaimEdit

The segment is equal to and starts at T.

ProofEdit

  1. Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.



  2. Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.



  3. equals to the sum of its parts and .



  4. equals to the sum of its parts and .



  5. The segment is equal to since they are the sides of the equilateral triangle .



  6. Since the sum of segments is equal and two of the summands are equal so are the two other summands and .



  7. Therefore equals .



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