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.
- Construct the equilateral triangle on .
- Bisect an angle on using the segment .
- Let C be the intersection point of and .
- Both and are equal to half of .
- 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.