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Thread: Clarification of proof

  1. #1
    Senior Member OReilly's Avatar
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    Clarification of proof

    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.
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  2. #2
    Grand Panjandrum
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    Quote 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.
    In your second paragraph you have to find a point $\displaystyle C_1$ on $\displaystyle l_1$ that satisfies
    two conditions, that is equivalent to finding a real number which satisfies two
    equations, there is no apriori guarantee that such a point exists. By
    introducing the point $\displaystyle B_2$ you guarantee the existence of both $\displaystyle C_1$ and $\displaystyle B_2$.

    Finally you show that the conditions imply that $\displaystyle B_2=B_1$.

    RonL
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