Bisection of Straight Line

Theorem
It is possible to bisect a straight line segment.

Construction


Let $$AB$$ be the given straight line segment.

We construct an equilateral triangle $$\triangle ABC$$ on $$AB$$.

We bisect the angle $$\angle ACB$$ by the straight line segment $$CD$$.

Then $$AB$$ has been bisected at the point $$D$$.

Proof
As $$\triangle ABC$$ is an equilateral triangle, it follows that $$AC = CB$$.

The two triangles $$\triangle ACD$$ and $$\triangle BCD$$ have side $$CD$$ in common, and side $$AC$$ of $$\triangle ACD$$ equals side $$BC$$ of $$\triangle BCD$$.

The angle $$\angle ACD$$ subtended by lines $$AC$$ and $$CD$$ equals the angle $$\angle BCD$$ subtended by lines $$BC$$ and $$CD$$, as $$\angle ACB$$ was bisected.

So triangles $\triangle ACD$ and $\triangle BCD$ are equal.

Therefore $$AD = DB$$.

So $$AB$$ has been bisected at the point $$D$$.

Note
This is Proposition 10 of Book I of Euclid's "The Elements".