Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff...












1












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This question already has an answer here:




  • Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$

    1 answer





A.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.

B.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$




I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.






complex-analysis complex-numbers







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marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, Namaste, José Carlos Santos complex-analysis
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 Jan 14 at 20:36


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


















  • $begingroup$
    You could start deriving B. from the assertion you already know?
    $endgroup$
    – Matteo
    Jan 11 at 15:20






  • 1




    $begingroup$
    I just found that I could prove B.
    $endgroup$
    – Jaqen Chou
    Jan 11 at 15:46










  • $begingroup$
    Be careful that your problem requires checking both necessary and sufficient condition.
    $endgroup$
    – Matteo
    Jan 11 at 16:14










  • $begingroup$
    Duplicate of math.stackexchange.com/q/921220
    $endgroup$
    – Jean Marie
    Jan 11 at 16:42
















1












$begingroup$



This question already has an answer here:




  • Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$

    1 answer





A.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.

B.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$




I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.






complex-analysis complex-numbers







share|cite|improve this question











$endgroup$



marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, Namaste, José Carlos Santos complex-analysis
Users with the  complex-analysis badge can single-handedly close complex-analysis questions as duplicates and reopen them as needed.

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 Jan 14 at 20:36


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


















  • $begingroup$
    You could start deriving B. from the assertion you already know?
    $endgroup$
    – Matteo
    Jan 11 at 15:20






  • 1




    $begingroup$
    I just found that I could prove B.
    $endgroup$
    – Jaqen Chou
    Jan 11 at 15:46










  • $begingroup$
    Be careful that your problem requires checking both necessary and sufficient condition.
    $endgroup$
    – Matteo
    Jan 11 at 16:14










  • $begingroup$
    Duplicate of math.stackexchange.com/q/921220
    $endgroup$
    – Jean Marie
    Jan 11 at 16:42














1












1








1


2



$begingroup$



This question already has an answer here:




  • Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$

    1 answer





A.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.

B.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$




I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.






complex-analysis complex-numbers







share|cite|improve this question











$endgroup$





This question already has an answer here:




  • Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$

    1 answer





A.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff
$bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1=r$ which $r$ is a real number.

B.

Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $lambda_1,lambda_2,lambda_3$ such that $lambda_1+lambda_2+lambda_3=0$ and $lambda_1z_1+lambda_2z_2+lambda_3z_3=0$




I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.





This question already has an answer here:




  • Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$

    1 answer







complex-analysis complex-numbers




complex-analysis complex-numbers






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 11 at 15:14









David C. Ullrich

61.2k43994




61.2k43994










asked Jan 11 at 15:12









Jaqen ChouJaqen Chou

451110




451110




marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, Namaste, José Carlos Santos complex-analysis
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 Jan 14 at 20:36


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Martin R, A. Pongrácz, Robert Soupe, Namaste, José Carlos Santos complex-analysis
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 Jan 14 at 20:36


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • $begingroup$
    You could start deriving B. from the assertion you already know?
    $endgroup$
    – Matteo
    Jan 11 at 15:20






  • 1




    $begingroup$
    I just found that I could prove B.
    $endgroup$
    – Jaqen Chou
    Jan 11 at 15:46










  • $begingroup$
    Be careful that your problem requires checking both necessary and sufficient condition.
    $endgroup$
    – Matteo
    Jan 11 at 16:14










  • $begingroup$
    Duplicate of math.stackexchange.com/q/921220
    $endgroup$
    – Jean Marie
    Jan 11 at 16:42


















  • $begingroup$
    You could start deriving B. from the assertion you already know?
    $endgroup$
    – Matteo
    Jan 11 at 15:20






  • 1




    $begingroup$
    I just found that I could prove B.
    $endgroup$
    – Jaqen Chou
    Jan 11 at 15:46










  • $begingroup$
    Be careful that your problem requires checking both necessary and sufficient condition.
    $endgroup$
    – Matteo
    Jan 11 at 16:14










  • $begingroup$
    Duplicate of math.stackexchange.com/q/921220
    $endgroup$
    – Jean Marie
    Jan 11 at 16:42
















$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20




$begingroup$
You could start deriving B. from the assertion you already know?
$endgroup$
– Matteo
Jan 11 at 15:20




1




1




$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46




$begingroup$
I just found that I could prove B.
$endgroup$
– Jaqen Chou
Jan 11 at 15:46












$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14




$begingroup$
Be careful that your problem requires checking both necessary and sufficient condition.
$endgroup$
– Matteo
Jan 11 at 16:14












$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42




$begingroup$
Duplicate of math.stackexchange.com/q/921220
$endgroup$
– Jean Marie
Jan 11 at 16:42










1 Answer
1






active

oldest

votes


















1












$begingroup$

I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :



Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is



$$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$



(see for example Equation of line in form of determinant)



Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :



$$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$



which, once developed, is equivalent to



$$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$



But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.






share|cite|improve this answer











$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :



    Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is



    $$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$



    (see for example Equation of line in form of determinant)



    Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :



    $$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$



    which, once developed, is equivalent to



    $$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$



    But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :



      Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is



      $$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$



      (see for example Equation of line in form of determinant)



      Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :



      $$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$



      which, once developed, is equivalent to



      $$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$



      But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :



        Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is



        $$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$



        (see for example Equation of line in form of determinant)



        Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :



        $$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$



        which, once developed, is equivalent to



        $$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$



        But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.






        share|cite|improve this answer











        $endgroup$



        I propose a solution that is different from the solution given in Why $bar{z_1}z_2+bar{z_2}z_3+bar{z_3}z_1 in mathbb R iff z_1, z_2, z_3 text{ are along the same line}$ :



        Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is



        $$begin{vmatrix}z_1&z_2&z\ overline{z_1} & overline{z_2}& overline{z}\1&1&1end{vmatrix}=0 ?$$



        (see for example Equation of line in form of determinant)



        Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :



        $$begin{vmatrix}z_1&z_2&z_3\ overline{z_1} & overline{z_2}& overline{z_3}\1&1&1end{vmatrix}=0tag{1}$$



        which, once developed, is equivalent to



        $$overline{z_2}z_1+overline{z_1}z_3+overline{z_3}z_2=overline{z_1}z_2+overline{z_3}z_1+overline{z_2}z_3 tag{2}$$



        But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 11 at 23:18

























        answered Jan 11 at 17:10









        Jean MarieJean Marie

        30.6k42154




        30.6k42154















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