# Geometry for Elementary School/Bisecting a segment

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In this chapter, we will learn how to bisect a segment. Given a segment , we will divide it to two equal segments and . The construction is based on book I, proposition 10.

## The construction

[edit | edit source]- Construct the equilateral triangle on .
- Bisect an angle on using the segment .
- Let
**C**be the intersection point of and .

## Claim

[edit | edit source]- Both and are equal to half of .

## The proof

[edit | edit source]- and are sides of the equilateral triangle .
- Hence, equals .
- The segment equals to itself.
- Due to the construction and are equal.
- The segments and lie on each other.
- Hence, equals to and equals to .
- Due to the Side-Angle-Side congruence theorem the triangles and congruent.
- Hence, and are equal.
- Since is the sum of and , each of them equals to its half.