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.

Geometria per scuola elementare/Bisezione di un segmento