Locus of centroid of triangle inside a circle.












3















A circle $x^2+y^2 = 3$ has a chord $AB$ which subtends an angle of $45^{circ}$ on a moving point $P$ on the circle. Find the locus of centroid of triangle $PAB$.







euclidean-geometry analytic-geometry geometric-transformation







share|cite|improve this question









New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – José Carlos Santos
    3 hours ago










  • Please phrase the question properly. If possible supply a diagram.
    – John Mitchell
    3 hours ago
















3















A circle $x^2+y^2 = 3$ has a chord $AB$ which subtends an angle of $45^{circ}$ on a moving point $P$ on the circle. Find the locus of centroid of triangle $PAB$.







euclidean-geometry analytic-geometry geometric-transformation







share|cite|improve this question









New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – José Carlos Santos
    3 hours ago










  • Please phrase the question properly. If possible supply a diagram.
    – John Mitchell
    3 hours ago














3












3








3


1






A circle $x^2+y^2 = 3$ has a chord $AB$ which subtends an angle of $45^{circ}$ on a moving point $P$ on the circle. Find the locus of centroid of triangle $PAB$.







euclidean-geometry analytic-geometry geometric-transformation







share|cite|improve this question









New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












A circle $x^2+y^2 = 3$ has a chord $AB$ which subtends an angle of $45^{circ}$ on a moving point $P$ on the circle. Find the locus of centroid of triangle $PAB$.







euclidean-geometry analytic-geometry geometric-transformation




euclidean-geometry analytic-geometry geometric-transformation






share|cite|improve this question









New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 1 hour ago









greedoid

37.2k114794




37.2k114794






New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 3 hours ago









Harsh Joshi

192




192




New contributor




Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Harsh Joshi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – José Carlos Santos
    3 hours ago










  • Please phrase the question properly. If possible supply a diagram.
    – John Mitchell
    3 hours ago


















  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
    – José Carlos Santos
    3 hours ago










  • Please phrase the question properly. If possible supply a diagram.
    – John Mitchell
    3 hours ago
















Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
3 hours ago




Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
3 hours ago












Please phrase the question properly. If possible supply a diagram.
– John Mitchell
3 hours ago




Please phrase the question properly. If possible supply a diagram.
– John Mitchell
3 hours ago










2 Answers
2






active

oldest

votes


















2














enter image description here



Radius of the circle is $R=sqrt3$. Introduce the following coordinates:



$A(-frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $B(frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $P(x_P,y_P)$.



Coordinates of the centroid are:



$$x_T=frac{x_A+x_B+x_P}{3}=frac {x_P}3$$



$$y_T=frac{y_A+y_B+y_P}{3}=frac {sqrt 6+y_P}3$$



or:



$$3 x_T={x_P}$$



$$3y_T-sqrt6=y_P$$



$$9 x_T^2+(3y_T-sqrt6)^2=x_P^2+y_P^2=3$$



$$9 x_T^2+9(y_T-frac{sqrt6}{3})^2=3$$



$$x_T^2+(y_T-frac{sqrt6}{3})^2=(frac{sqrt3}3)^2$$



...which is a circle with center $(0, frac{sqrt{6}}{3})$ and radius $frac{sqrt3}3$






share|cite|improve this answer





























    0














    Anyway, this could be done also with synthetic (Euclidian) geometry.



    Let $S= (0,0)$ be a center of a given circle.



    Let $M$ be a midpoint of a chord $AB$. Then $vec{MT} = {1over 3}vec{MP}$ for every $P$. So we get $T$ after performing a homothety with center at $M$ and dilatation coefficient $k= {1over 3}$ on $P$. Since $P$ is on a fixed circle and homothety takes a circle to a circle we see that $T$ describe a circle with center at $C$ which is on ${1over 3}MS$ and with radius $r' = {rover 3} = {sqrt{3}over 3}$.



    So if we take a picture of oldboy we have $M=(0,{sqrt{6}over 2})$ so $C = (0,{sqrt{6}over 3})$ and thus $$x^2+(y-{sqrt{6}over 3})^2= {1over 3}$$






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });






      Harsh Joshi is a new contributor. Be nice, and check out our Code of Conduct.










      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051951%2flocus-of-centroid-of-triangle-inside-a-circle%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2














      enter image description here



      Radius of the circle is $R=sqrt3$. Introduce the following coordinates:



      $A(-frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $B(frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $P(x_P,y_P)$.



      Coordinates of the centroid are:



      $$x_T=frac{x_A+x_B+x_P}{3}=frac {x_P}3$$



      $$y_T=frac{y_A+y_B+y_P}{3}=frac {sqrt 6+y_P}3$$



      or:



      $$3 x_T={x_P}$$



      $$3y_T-sqrt6=y_P$$



      $$9 x_T^2+(3y_T-sqrt6)^2=x_P^2+y_P^2=3$$



      $$9 x_T^2+9(y_T-frac{sqrt6}{3})^2=3$$



      $$x_T^2+(y_T-frac{sqrt6}{3})^2=(frac{sqrt3}3)^2$$



      ...which is a circle with center $(0, frac{sqrt{6}}{3})$ and radius $frac{sqrt3}3$






      share|cite|improve this answer


























        2














        enter image description here



        Radius of the circle is $R=sqrt3$. Introduce the following coordinates:



        $A(-frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $B(frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $P(x_P,y_P)$.



        Coordinates of the centroid are:



        $$x_T=frac{x_A+x_B+x_P}{3}=frac {x_P}3$$



        $$y_T=frac{y_A+y_B+y_P}{3}=frac {sqrt 6+y_P}3$$



        or:



        $$3 x_T={x_P}$$



        $$3y_T-sqrt6=y_P$$



        $$9 x_T^2+(3y_T-sqrt6)^2=x_P^2+y_P^2=3$$



        $$9 x_T^2+9(y_T-frac{sqrt6}{3})^2=3$$



        $$x_T^2+(y_T-frac{sqrt6}{3})^2=(frac{sqrt3}3)^2$$



        ...which is a circle with center $(0, frac{sqrt{6}}{3})$ and radius $frac{sqrt3}3$






        share|cite|improve this answer
























          2












          2








          2






          enter image description here



          Radius of the circle is $R=sqrt3$. Introduce the following coordinates:



          $A(-frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $B(frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $P(x_P,y_P)$.



          Coordinates of the centroid are:



          $$x_T=frac{x_A+x_B+x_P}{3}=frac {x_P}3$$



          $$y_T=frac{y_A+y_B+y_P}{3}=frac {sqrt 6+y_P}3$$



          or:



          $$3 x_T={x_P}$$



          $$3y_T-sqrt6=y_P$$



          $$9 x_T^2+(3y_T-sqrt6)^2=x_P^2+y_P^2=3$$



          $$9 x_T^2+9(y_T-frac{sqrt6}{3})^2=3$$



          $$x_T^2+(y_T-frac{sqrt6}{3})^2=(frac{sqrt3}3)^2$$



          ...which is a circle with center $(0, frac{sqrt{6}}{3})$ and radius $frac{sqrt3}3$






          share|cite|improve this answer












          enter image description here



          Radius of the circle is $R=sqrt3$. Introduce the following coordinates:



          $A(-frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $B(frac{sqrt{6}}{2}, frac{sqrt{6}}{2})$, $P(x_P,y_P)$.



          Coordinates of the centroid are:



          $$x_T=frac{x_A+x_B+x_P}{3}=frac {x_P}3$$



          $$y_T=frac{y_A+y_B+y_P}{3}=frac {sqrt 6+y_P}3$$



          or:



          $$3 x_T={x_P}$$



          $$3y_T-sqrt6=y_P$$



          $$9 x_T^2+(3y_T-sqrt6)^2=x_P^2+y_P^2=3$$



          $$9 x_T^2+9(y_T-frac{sqrt6}{3})^2=3$$



          $$x_T^2+(y_T-frac{sqrt6}{3})^2=(frac{sqrt3}3)^2$$



          ...which is a circle with center $(0, frac{sqrt{6}}{3})$ and radius $frac{sqrt3}3$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          Oldboy

          6,5401730




          6,5401730























              0














              Anyway, this could be done also with synthetic (Euclidian) geometry.



              Let $S= (0,0)$ be a center of a given circle.



              Let $M$ be a midpoint of a chord $AB$. Then $vec{MT} = {1over 3}vec{MP}$ for every $P$. So we get $T$ after performing a homothety with center at $M$ and dilatation coefficient $k= {1over 3}$ on $P$. Since $P$ is on a fixed circle and homothety takes a circle to a circle we see that $T$ describe a circle with center at $C$ which is on ${1over 3}MS$ and with radius $r' = {rover 3} = {sqrt{3}over 3}$.



              So if we take a picture of oldboy we have $M=(0,{sqrt{6}over 2})$ so $C = (0,{sqrt{6}over 3})$ and thus $$x^2+(y-{sqrt{6}over 3})^2= {1over 3}$$






              share|cite|improve this answer


























                0














                Anyway, this could be done also with synthetic (Euclidian) geometry.



                Let $S= (0,0)$ be a center of a given circle.



                Let $M$ be a midpoint of a chord $AB$. Then $vec{MT} = {1over 3}vec{MP}$ for every $P$. So we get $T$ after performing a homothety with center at $M$ and dilatation coefficient $k= {1over 3}$ on $P$. Since $P$ is on a fixed circle and homothety takes a circle to a circle we see that $T$ describe a circle with center at $C$ which is on ${1over 3}MS$ and with radius $r' = {rover 3} = {sqrt{3}over 3}$.



                So if we take a picture of oldboy we have $M=(0,{sqrt{6}over 2})$ so $C = (0,{sqrt{6}over 3})$ and thus $$x^2+(y-{sqrt{6}over 3})^2= {1over 3}$$






                share|cite|improve this answer
























                  0












                  0








                  0






                  Anyway, this could be done also with synthetic (Euclidian) geometry.



                  Let $S= (0,0)$ be a center of a given circle.



                  Let $M$ be a midpoint of a chord $AB$. Then $vec{MT} = {1over 3}vec{MP}$ for every $P$. So we get $T$ after performing a homothety with center at $M$ and dilatation coefficient $k= {1over 3}$ on $P$. Since $P$ is on a fixed circle and homothety takes a circle to a circle we see that $T$ describe a circle with center at $C$ which is on ${1over 3}MS$ and with radius $r' = {rover 3} = {sqrt{3}over 3}$.



                  So if we take a picture of oldboy we have $M=(0,{sqrt{6}over 2})$ so $C = (0,{sqrt{6}over 3})$ and thus $$x^2+(y-{sqrt{6}over 3})^2= {1over 3}$$






                  share|cite|improve this answer












                  Anyway, this could be done also with synthetic (Euclidian) geometry.



                  Let $S= (0,0)$ be a center of a given circle.



                  Let $M$ be a midpoint of a chord $AB$. Then $vec{MT} = {1over 3}vec{MP}$ for every $P$. So we get $T$ after performing a homothety with center at $M$ and dilatation coefficient $k= {1over 3}$ on $P$. Since $P$ is on a fixed circle and homothety takes a circle to a circle we see that $T$ describe a circle with center at $C$ which is on ${1over 3}MS$ and with radius $r' = {rover 3} = {sqrt{3}over 3}$.



                  So if we take a picture of oldboy we have $M=(0,{sqrt{6}over 2})$ so $C = (0,{sqrt{6}over 3})$ and thus $$x^2+(y-{sqrt{6}over 3})^2= {1over 3}$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  greedoid

                  37.2k114794




                  37.2k114794






















                      Harsh Joshi is a new contributor. Be nice, and check out our Code of Conduct.










                      draft saved

                      draft discarded


















                      Harsh Joshi is a new contributor. Be nice, and check out our Code of Conduct.













                      Harsh Joshi is a new contributor. Be nice, and check out our Code of Conduct.












                      Harsh Joshi is a new contributor. Be nice, and check out our Code of Conduct.
















                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051951%2flocus-of-centroid-of-triangle-inside-a-circle%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      404 Error Contact Form 7 ajax form submitting

                      Image
                      0 The contact form does not work, and affects loading after one try. The chrome console shows the error message: POST http://example.com/wp-json/contact-form-7/v1/contact-forms/50/feedback 404 (Not Found) send @ jquery.js?ver=1.12.4:4 ajax @ jquery.js?ver=1.12.4:4 wpcf7.submit @ scripts.js?ver=5.0.3:346 (anonymous) @ scripts.js?ver=5.0.3:53 dispatch @ jquery.js?ver=1.12.4:3 r.handle @ jquery.js?ver=1.12.4:3 How to solve this issue? thanks in advance javascript wordpress contact-form-7 share | improve this question edited Jul 17 '18 at 5:39 Kiran Shahi 4,745 3 16 43 ...
                      Read more

                      How to know if a Active Directory user can login interactively

                      Image
                      1 I would like to know if and how is it possible to know if a AD user can login interactively (on a server) in a domain. I need to know if I can find it out using a LDAP search. Thanks. windows active-directory login user-permissions user-profile share | improve this question edited 1 hour ago yagmoth555 ♦ 11.6k 3 17 42 asked 1 hour ago Luigi Luigi 6 1 New contributor ...
                      Read more

                      Refactoring coordinates for Minecraft Pi buildings written in Python

                      Image
                      0 $begingroup$ I'm fairly new to coding and I wrote this script in Python for a house with a roof in Minecraft Pi. I've found different ways to write the coordinates to make the code base shorter, but I'm wondering if it is possible for me to refactor it any further so I don't have to list each coordinate line by line. import mcpi.minecraft as minecraft import mcpi.block as block mc = minecraft.Minecraft.create() p = mc.player.getTilePos() mc.setBlocks(p.x+1, p.y, p.z+1, #Main structure starting coordinates (DO NOT ALTER) p.x+19, p.y+4, p.z+6, #Structure size/coordinates (Alters cube size) block.STONE_BRICK) mc.setBlocks(p.x+2, p.y+1, p.z+2, #Hollowing out main structure interior p.x+18, p.y+3, p.z+5, block.AIR) mc.setBlocks(p.x, p.y...
                      Read more