# Geometry for Elementary School/Copying a line segment

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

## The construction[edit | edit source]

- 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**). - Draw a line segment
- Construct an equilateral triangle (a triangle that has as one of its sides).
- Draw the circle , whose center is
**A**and radius is . - Draw a line segment starting from
**D**going through**A**until it intersects and let the intersection point be**E**. Get segments and . - Draw the circle , whose center is
**D**and radius is . - Draw a line segment starting from
**D**going through**T**until it intersects and let the intersection point be**F**. Get segments and .

## Claim[edit | edit source]

The segment is equal to and starts at **T**.

## Proof[edit | edit source]

- Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.
- Segments and are both from the center of to its circumference. Therefore they equal to the circle radius and to each other.
- equals to the sum of its parts and .
- equals to the sum of its parts and .
- The segment is equal to since they are the sides of the equilateral triangle .
- Since the sum of segments is equal and two of the summands are equal so are the two other summands and .
- Therefore equals .