Two Lines Meet at Unique Point

Theorem
Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point.

Then there can not be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point.

Proof


Suppose $$AC$$ and $$CB$$ have been constructed on $$AB$$ meeting at $$C$$.

Let two other straight line segments $$AD$$ and $$DB$$ be constructed on $$AB$$, on the same side of it, meeting at $$D$$.

Let $$CD$$ be joined.

Since $$AC = AD$$ it follows that $\angle ACD = \angle ADC$.

Therefore $$\angle ACD$$ is greater than $$\angle DCB$$ because the whole is greater than the part.

Therefore $$\angle CDB$$ is much greater than $$\angle DCB$$.

Now since $$CD = DB$$, it follows that $$\angle CDB = \angle DCB$$.

But it was proved much greater than it.

From this contradiction we conclude the result.

Note
This is Proposition 7 of Book I of Euclid's "The Elements".