# Geometry for Elementary School/Bisecting a segment

 Geometry for Elementary School Bisecting an angle Bisecting a segment Congruence and similarity

In this chapter, we will learn how to bisect a segment. Given a segment ${\overline {AB}}$ , we will divide it to two equal segments ${\overline {AC}}$ and ${\overline {CB}}$ . The construction is based on book I, proposition 10.

## The construction

1. Construct the equilateral triangle $\triangle ABD$  on ${\overline {AB}}$ .
2. Bisect an angle on $\angle ADB$  using the segment ${\overline {DE}}$ .
3. Let C be the intersection point of ${\overline {DE}}$  and ${\overline {AB}}$ .

## Claim

1. Both ${\overline {AC}}$  and ${\overline {CB}}$  are equal to half of ${\overline {AB}}$ .

## The proof

1. ${\overline {AD}}$  and ${\overline {BD}}$  are sides of the equilateral triangle $\triangle ABD$ .
2. Hence, ${\overline {AD}}$  equals ${\overline {BD}}$ .
3. The segment ${\overline {DC}}$  equals to itself.
4. Due to the construction $\angle ADE$  and $\angle EDB$  are equal.
5. The segments ${\overline {DE}}$  and ${\overline {DC}}$  lie on each other.
6. Hence, $\angle ADE$  equals to $\angle ADC$  and $\angle EDB$  equals to $\angle CDB$ .
7. Due to the Side-Angle-Side congruence theorem the triangles $\triangle ADC$  and $\triangle CDB$  congruent.
8. Hence, ${\overline {AC}}$  and ${\overline {CB}}$  are equal.
9. Since ${\overline {AB}}$  is the sum of ${\overline {AC}}$  and ${\overline {CB}}$ , each of them equals to its half.