Geometry for Elementary School/Bisecting a segment

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.