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Math Help - 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 A,B,C are tri different points of line l and A_1,B_1,C_1 are points of line l_1 such that (A,B) \cong (A_1 ,B_1 ), then there exists unique point C_1 such that (A,C) \cong (A_1 ,C_1 ) and (B,C) \cong (B_1 ,C_1 ). Also, point C_1 belongs to line l_1 and ordering of points A,B,C on line l matches ordering of points A_1,B_1,C_1 on line l_1.


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


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



    I don't understand why we need point B_2 at all. If we asume that there is point C_1 such that A_1-C_1-B_1 , (A,C) \cong (A_1 ,C_1 ) and (B,C) \cong (B_1 ,C_1 ) then it follows (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 A,B,C are tri different points of line l and A_1,B_1,C_1 are points of line l_1 such that (A,B) \cong (A_1 ,B_1 ), then there exists unique point C_1 such that (A,C) \cong (A_1 ,C_1 ) and (B,C) \cong (B_1 ,C_1 ). Also, point C_1 belongs to line l_1 and ordering of points A,B,C on line l matches ordering of points A_1,B_1,C_1 on line l_1.


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


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



    I don't understand why we need point B_2 at all. If we asume that there is point C_1 such that A_1-C_1-B_1 , (A,C) \cong (A_1 ,C_1 ) and (B,C) \cong (B_1 ,C_1 ) then it follows (A,B) \cong (A_1 ,B_1 ) which is the same proof.
    In your second paragraph you have to find a point C_1 on 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 B_2 you guarantee the existence of both C_1 and B_2.

    Finally you show that the conditions imply that B_2=B_1.

    RonL
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