Originally Posted by **OReilly**

Can someone clarify me this proof for this theorem?

**Theorem:** If $\displaystyle A,B,C$ are tri different points of line $\displaystyle l$ and $\displaystyle A_1,B_1,C_1$ are points of line $\displaystyle l_1$ such that $\displaystyle (A,B) \cong (A_1 ,B_1 )$, then there exists unique point $\displaystyle C_1$ such that $\displaystyle (A,C) \cong (A_1 ,C_1 )$ and $\displaystyle (B,C) \cong (B_1 ,C_1 )$. Also, point $\displaystyle C_1$ belongs to line $\displaystyle l_1$ and ordering of points $\displaystyle A,B,C$ on line $\displaystyle l$ matches ordering of points $\displaystyle A_1,B_1,C_1$ on line $\displaystyle l_1$.

I will show proof for order of points $\displaystyle A-C-B$.

**Proof: ** If $\displaystyle C_1$ and $\displaystyle B_2$ are points of ray $\displaystyle A_1B_1$ such that $\displaystyle A_1-C_1-B_2$, $\displaystyle (A,C) \cong (A_1 ,C_1 )$ and $\displaystyle (B,C) \cong (B_2 ,C_1 )$ then it follows $\displaystyle (A,B) \cong (A_1 ,B_2 )$ so because of $\displaystyle B_1=B_2$ there is point $\displaystyle C_1$ that meets conditions of theorem.

I don't understand why we need point $\displaystyle B_2$ at all. If we asume that there is point $\displaystyle C_1$ such that $\displaystyle A_1-C_1-B_1$ ,$\displaystyle (A,C) \cong (A_1 ,C_1 )$ and $\displaystyle (B,C) \cong (B_1 ,C_1 )$ then it follows $\displaystyle (A,B) \cong (A_1 ,B_1 )$ which is the same proof.