Last modified on 15 April 2014, at 20:13

Geometry for Elementary School/Copying a line segment

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


This construction copies a line segment \overline{AB} 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 \overline{AB}. 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 \overline{AT}
    Geom copyseg 02.png


  3. Construct an equilateral triangle \triangle ATD (a triangle that has \overline{AT} as one of its sides).
    Geom copyseg 03.png


  4. Draw the circle \circ A,\overline{AB} , whose center is A and radius is \overline{AB}.
    Geom copyseg 04.png


  5. Draw a line segment starting from D going through A until it intersects \circ A,\overline{AB} and let the intersection point be E . Get segments \overline{AE} and \overline{DE}.
    Geom copyseg 05.png


  6. Draw the circle \circ D,\overline{DE} , whose center is D and radius is \overline{DE}.
    Geom copyseg 06.png


  7. Draw a line segment starting from D going through T until it intersects \circ D,\overline{DE} and let the intersection point be F. Get segments \overline{TF} and \overline{DF}.
    Geom copyseg 07.png

ClaimEdit

The segment \overline{TF} is equal to \overline{AB} and starts at T.
Geom copyseg claim.png

ProofEdit

  1. Segments \overline{AB} and \overline{AE} are both from the center of \circ A,\overline{AB} to its circumference. Therefore they equal to the circle radius and to each other.
    Geom copyseg proof01.png


  2. Segments \overline{DE} and \overline{DF} are both from the center of \circ D,\overline{DE} to its circumference. Therefore they equal to the circle radius and to each other.
    Geom copyseg proof02.png


  3. \overline{DE} equals to the sum of its parts \overline{DA} and \overline{AE}.
    Geom copyseg proof03.png


  4. \overline{DF} equals to the sum of its parts \overline{DT} and \overline{TF}.
    Geom copyseg proof04.png


  5. The segment \overline{DA} is equal to \overline{DT} since they are the sides of the equilateral triangle \triangle ATD .
    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 \overline{AE} and \overline{TF}.
    Geom copyseg proof06.png


  7. Therefore \overline{AB} equals \overline{TF}.
    Geom copyseg proof07.png