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 , we will divide it to two equal segments and . The construction is based on book I, proposition 10.

The construction

  1. Construct the equilateral triangle   on  .
  2. Bisect an angle on   using the segment  .
  3. Let C be the intersection point of   and  .


  1. Both   and   are equal to half of  .

The proof

  1.   and   are sides of the equilateral triangle  .
  2. Hence,   equals  .
  3. The segment   equals to itself.
  4. Due to the construction   and   are equal.
  5. The segments   and   lie on each other.
  6. Hence,   equals to   and   equals to  .
  7. Due to the Side-Angle-Side congruence theorem the triangles   and   congruent.
  8. Hence,   and   are equal.
  9. Since   is the sum of   and  , each of them equals to its half.