Geometry for Elementary School/Bisecting a segment

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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[edit]

  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[edit]

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

The proof[edit]

  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.