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

The construction

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


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  1. Both and are equal to half of .

The proof

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  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.