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 construction[edit]

  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).
    Geom copyseg 01.png


  2. Draw a line segment
    Geom copyseg 02.png


  3. Construct an equilateral triangle (a triangle that has as one of its sides).
    Geom copyseg 03.png


  4. Draw the circle , whose center is A and radius is .
    Geom copyseg 04.png


  5. Draw a line segment starting from D going through A until it intersects and let the intersection point be E . Get segments and .
    Geom copyseg 05.png


  6. Draw the circle , whose center is D and radius is .
    Geom copyseg 06.png


  7. Draw a line segment starting from D going through T until it intersects and let the intersection point be F. Get segments and .
    Geom copyseg 07.png

Claim[edit]

The segment is equal to and starts at T.
Geom copyseg claim.png

Proof[edit]

  1. Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.
    Geom copyseg proof01.png


  2. Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.
    Geom copyseg proof02.png


  3. equals to the sum of its parts and .
    Geom copyseg proof03.png


  4. equals to the sum of its parts and .
    Geom copyseg proof04.png


  5. The segment is equal to since they are the sides of the equilateral triangle .
    Geom copyseg proof05.png


  6. Since the sum of segments is equal and two of the summands are equal so are the two other summands and .
    Geom copyseg proof06.png


  7. Therefore equals .
    Geom copyseg proof07.png