Geometry for Elementary School/Bisecting a segment

< Geometry for Elementary School

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

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

Claim

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

The proof

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

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