If $|MA|^2=|MB|^2+|MC|^2$. Find the $angle BMC=?$ [closed]












2












$begingroup$


My problem is:




The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$




Is this question related to similar triangles? Or which theorem should I apply?










share|cite|improve this question











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closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.





















    2












    $begingroup$


    My problem is:




    The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$




    Is this question related to similar triangles? Or which theorem should I apply?










    share|cite|improve this question











    $endgroup$



    closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo

    If this question can be reworded to fit the rules in the help center, please edit the question.



















      2












      2








      2





      $begingroup$


      My problem is:




      The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$




      Is this question related to similar triangles? Or which theorem should I apply?










      share|cite|improve this question











      $endgroup$




      My problem is:




      The point $M$ is the internal point of the $ABC$ equilateral triangle. Find the $angle BMC$ if $$|MA|^2=|MB|^2+|MC|^2$$




      Is this question related to similar triangles? Or which theorem should I apply?







      geometry contest-math euclidean-geometry book-recommendation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 26 '18 at 11:56









      greedoid

      44k1155109




      44k1155109










      asked Dec 26 '18 at 11:39









      ElementaryElementary

      361111




      361111




      closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by RRL, Lee David Chung Lin, jgon, KReiser, Cesareo Jan 5 at 0:29


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, Lee David Chung Lin, jgon, KReiser, Cesareo

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          3 Answers
          3






          active

          oldest

          votes


















          4












          $begingroup$

          Hint:



          Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?



          enter image description here



          Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I'm practicing the hint you're saying.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:56










          • $begingroup$
            Is it possible, to add picture, if you have a few time ..I could not do..:(
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:47










          • $begingroup$
            Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:00










          • $begingroup$
            I made some edit...
            $endgroup$
            – greedoid
            Dec 26 '18 at 13:03










          • $begingroup$
            Thank you again..
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:26



















          2












          $begingroup$

          Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
          $$|MA|^2=|MB|^2+|MC|^2$$
          $$implies y^2+2sqrt3ay-a^2=0$$
          $$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative



          Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How do you get, $sqrt3 a$
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:05












          • $begingroup$
            Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 12:06



















          1












          $begingroup$

          That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No, this is wrong approach.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:51






          • 1




            $begingroup$
            The answer is incorrect too
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 11:52










          • $begingroup$
            But why doesn't that work?
            $endgroup$
            – Viktor Glombik
            Dec 26 '18 at 12:39


















          3 Answers
          3






          active

          oldest

          votes








          3 Answers
          3






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4












          $begingroup$

          Hint:



          Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?



          enter image description here



          Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I'm practicing the hint you're saying.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:56










          • $begingroup$
            Is it possible, to add picture, if you have a few time ..I could not do..:(
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:47










          • $begingroup$
            Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:00










          • $begingroup$
            I made some edit...
            $endgroup$
            – greedoid
            Dec 26 '18 at 13:03










          • $begingroup$
            Thank you again..
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:26
















          4












          $begingroup$

          Hint:



          Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?



          enter image description here



          Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            I'm practicing the hint you're saying.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:56










          • $begingroup$
            Is it possible, to add picture, if you have a few time ..I could not do..:(
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:47










          • $begingroup$
            Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:00










          • $begingroup$
            I made some edit...
            $endgroup$
            – greedoid
            Dec 26 '18 at 13:03










          • $begingroup$
            Thank you again..
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:26














          4












          4








          4





          $begingroup$

          Hint:



          Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?



          enter image description here



          Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...






          share|cite|improve this answer











          $endgroup$



          Hint:



          Rotate $C$ around $B$ for $60^{circ}$ and let $N$ be a picture of $M$ under this rotation. What can you say for triangles $BMN$ and $AMN$?



          enter image description here



          Edit: Triangle $BMC$ must be equilateral and since $$ AN^2+MN^2 = CM^2+BM^2 = AM^2$$ we see that triangle $ANM$ is rectangular with 90 at $N$...







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 26 '18 at 13:03

























          answered Dec 26 '18 at 11:53









          greedoidgreedoid

          44k1155109




          44k1155109












          • $begingroup$
            I'm practicing the hint you're saying.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:56










          • $begingroup$
            Is it possible, to add picture, if you have a few time ..I could not do..:(
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:47










          • $begingroup$
            Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:00










          • $begingroup$
            I made some edit...
            $endgroup$
            – greedoid
            Dec 26 '18 at 13:03










          • $begingroup$
            Thank you again..
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:26


















          • $begingroup$
            I'm practicing the hint you're saying.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:56










          • $begingroup$
            Is it possible, to add picture, if you have a few time ..I could not do..:(
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:47










          • $begingroup$
            Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:00










          • $begingroup$
            I made some edit...
            $endgroup$
            – greedoid
            Dec 26 '18 at 13:03










          • $begingroup$
            Thank you again..
            $endgroup$
            – Elementary
            Dec 26 '18 at 13:26
















          $begingroup$
          I'm practicing the hint you're saying.
          $endgroup$
          – Elementary
          Dec 26 '18 at 11:56




          $begingroup$
          I'm practicing the hint you're saying.
          $endgroup$
          – Elementary
          Dec 26 '18 at 11:56












          $begingroup$
          Is it possible, to add picture, if you have a few time ..I could not do..:(
          $endgroup$
          – Elementary
          Dec 26 '18 at 12:47




          $begingroup$
          Is it possible, to add picture, if you have a few time ..I could not do..:(
          $endgroup$
          – Elementary
          Dec 26 '18 at 12:47












          $begingroup$
          Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
          $endgroup$
          – Elementary
          Dec 26 '18 at 13:00




          $begingroup$
          Thank you very much. Your method has exceeded my understanding.I could not. There is no problem in the solution. The problem is with me.
          $endgroup$
          – Elementary
          Dec 26 '18 at 13:00












          $begingroup$
          I made some edit...
          $endgroup$
          – greedoid
          Dec 26 '18 at 13:03




          $begingroup$
          I made some edit...
          $endgroup$
          – greedoid
          Dec 26 '18 at 13:03












          $begingroup$
          Thank you again..
          $endgroup$
          – Elementary
          Dec 26 '18 at 13:26




          $begingroup$
          Thank you again..
          $endgroup$
          – Elementary
          Dec 26 '18 at 13:26











          2












          $begingroup$

          Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
          $$|MA|^2=|MB|^2+|MC|^2$$
          $$implies y^2+2sqrt3ay-a^2=0$$
          $$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative



          Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How do you get, $sqrt3 a$
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:05












          • $begingroup$
            Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 12:06
















          2












          $begingroup$

          Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
          $$|MA|^2=|MB|^2+|MC|^2$$
          $$implies y^2+2sqrt3ay-a^2=0$$
          $$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative



          Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How do you get, $sqrt3 a$
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:05












          • $begingroup$
            Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 12:06














          2












          2








          2





          $begingroup$

          Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
          $$|MA|^2=|MB|^2+|MC|^2$$
          $$implies y^2+2sqrt3ay-a^2=0$$
          $$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative



          Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)






          share|cite|improve this answer









          $endgroup$



          Use coordinate geometry. Note that $|MA|^2=|MB|^2+|MC|^2$ is symmetrical in $B & C$. This means $M$ is located symmetrically with respect to $B & C$. Let the coordinates of $A$ be $(0,sqrt{3}a)$, $B$ be $(-a,0)$ and $C$ be $(a,0)$. Since $M$ has to be symmetrical, let the coordinates for $M$ be $(0,y)$. Now,
          $$|MA|^2=|MB|^2+|MC|^2$$
          $$implies y^2+2sqrt3ay-a^2=0$$
          $$implies y=(2-sqrt3)a$$Note that you'll get two values of y but it can't be negative



          Now since you've $M$, you can get the angle $BMC$ to be equal to $150$ degrees. I hope you can get there :)







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 26 '18 at 11:57









          Ankit KumarAnkit Kumar

          1,494221




          1,494221












          • $begingroup$
            How do you get, $sqrt3 a$
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:05












          • $begingroup$
            Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 12:06


















          • $begingroup$
            How do you get, $sqrt3 a$
            $endgroup$
            – Elementary
            Dec 26 '18 at 12:05












          • $begingroup$
            Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 12:06
















          $begingroup$
          How do you get, $sqrt3 a$
          $endgroup$
          – Elementary
          Dec 26 '18 at 12:05






          $begingroup$
          How do you get, $sqrt3 a$
          $endgroup$
          – Elementary
          Dec 26 '18 at 12:05














          $begingroup$
          Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
          $endgroup$
          – Ankit Kumar
          Dec 26 '18 at 12:06




          $begingroup$
          Since it's an equilateral triangle, I've taken the length of each side to be $2a$ (calculations will be a little bit easy with 2a instead of just a). You know how I got $sqrt3a$ now right?
          $endgroup$
          – Ankit Kumar
          Dec 26 '18 at 12:06











          1












          $begingroup$

          That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No, this is wrong approach.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:51






          • 1




            $begingroup$
            The answer is incorrect too
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 11:52










          • $begingroup$
            But why doesn't that work?
            $endgroup$
            – Viktor Glombik
            Dec 26 '18 at 12:39
















          1












          $begingroup$

          That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No, this is wrong approach.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:51






          • 1




            $begingroup$
            The answer is incorrect too
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 11:52










          • $begingroup$
            But why doesn't that work?
            $endgroup$
            – Viktor Glombik
            Dec 26 '18 at 12:39














          1












          1








          1





          $begingroup$

          That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.






          share|cite|improve this answer









          $endgroup$



          That equation is essentially the Pythagorean theorem applied to lines MA, MB, MC. Even though they do not form a triangle in this scenario, the equation shows that they can. Line MA is the hypotenuse in this case, which makes angle BMC 90 degrees.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 26 '18 at 11:48









          FrostFrost

          1776




          1776








          • 1




            $begingroup$
            No, this is wrong approach.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:51






          • 1




            $begingroup$
            The answer is incorrect too
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 11:52










          • $begingroup$
            But why doesn't that work?
            $endgroup$
            – Viktor Glombik
            Dec 26 '18 at 12:39














          • 1




            $begingroup$
            No, this is wrong approach.
            $endgroup$
            – Elementary
            Dec 26 '18 at 11:51






          • 1




            $begingroup$
            The answer is incorrect too
            $endgroup$
            – Ankit Kumar
            Dec 26 '18 at 11:52










          • $begingroup$
            But why doesn't that work?
            $endgroup$
            – Viktor Glombik
            Dec 26 '18 at 12:39








          1




          1




          $begingroup$
          No, this is wrong approach.
          $endgroup$
          – Elementary
          Dec 26 '18 at 11:51




          $begingroup$
          No, this is wrong approach.
          $endgroup$
          – Elementary
          Dec 26 '18 at 11:51




          1




          1




          $begingroup$
          The answer is incorrect too
          $endgroup$
          – Ankit Kumar
          Dec 26 '18 at 11:52




          $begingroup$
          The answer is incorrect too
          $endgroup$
          – Ankit Kumar
          Dec 26 '18 at 11:52












          $begingroup$
          But why doesn't that work?
          $endgroup$
          – Viktor Glombik
          Dec 26 '18 at 12:39




          $begingroup$
          But why doesn't that work?
          $endgroup$
          – Viktor Glombik
          Dec 26 '18 at 12:39



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