Geometry for Elementary School/Copying a line segment

This construction copies a line segment $$\overline{AB}$$ to a target point T. The construction is based on Book I, prop 2.

The construction

 * 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 $$\overline{AT}$$ [[image:Geom copyseg 02.png|225px]]
 * 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

Claim
The segment $$\overline{TF}$$ is equal to $$\overline{AB}$$ and starts at T.

Proof

 * 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

Geometria per scuola elementare/Copia di un segmento