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  .