## Bisecting a Line

**How to Bisect a Line**

**The Theory Behind How it Works**

**Bisecting a line**means that we want to divide that given line into 2 equal parts.

From the image given above we want to divide line AB in to 2 equal parts.

**Proof:**

- Lines AC, AD, BC, and BD are all congruent because they were all drawn with the same radius, radius E.

Next we prove that the top and bottom triangles are isosceles and congruent. (See Chapter on triangle, Parts of triangles)

- Triangles ∆ABC and ∆ABD are congruent and isosceles because as we already know AC, AD, BC and BD are all equal as they were all drawn with the same radius, radius E and line CD is common to both triangles.
- Angles CAF and CBF are congruent because the base angles in an isosceles triangle are always equal.
- Angles CAF, CBF, DAF, and DBF are congruent (the angles with the solid black dot) because since they are congruent isosceles triangles, the base angles are all equal.

Next we prove that the left and right triangles are isosceles triangles are isosceles and congruent.

- Triangles ∆ADC and ∆BDC are congruent and isosceles because as we already know AC, AD, BC and BD are all equal as they were all drawn with the same radius, radius E and line AB is common to both triangles.
- Angles ADF and ACF are congruent because the base angles in an isosceles triangle are always equal.
- Angles ADF, ACF, BCF, and BDF are congruent (the angles with the 'O' symbol) because since they are congruent isosceles triangles, the base angles are always equal.

Finally we prove that the 4 smaller triangles are congruent and finish the proof.

- Triangles ∆ACF, ∆BCF, ∆ADF and ∆BDF are congruent because they have 2 angles and a side (length E) in common.
- The 4 angles at F, AFC, AFD, BFC and BFD are congruent because from step 1 in this section we know that the 4 triangles have 2 angles in common, therefore the remaining 4 angles which are at F must be equal.
- Therefore the 4 angles at F must be 90° as they must add up to 360°. Consequently line CD is perpendicular to AB.
- Hence line AF and BF are congruent, therefore line CD bisects line AB into 2 equal parts

Q.E.D.

**Bisecting a Line Interactive**