Find the locus of the centroid of the triangle?












0















A plane moves so that its distance from the origin is a constant $p$. Write down the equation of the locus of the centroid of the triangle formed by its intersection with the three coordinates plane.




I was thinking about this question many times, but i could not get any hint to find the locus of the centroid of the triangle .
But i know that how to find the centroid of triangle we have to add (x+y+z/3,X+Y+Z/3), i don't know the locus of the centroid of the triangle.



if anbody help me i would be very thankful to him






geometry analytic-geometry locus







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  • So the plane is always tangent to the sphere?
    – Michael Hoppe
    Aug 9 '17 at 5:53


















0















A plane moves so that its distance from the origin is a constant $p$. Write down the equation of the locus of the centroid of the triangle formed by its intersection with the three coordinates plane.




I was thinking about this question many times, but i could not get any hint to find the locus of the centroid of the triangle .
But i know that how to find the centroid of triangle we have to add (x+y+z/3,X+Y+Z/3), i don't know the locus of the centroid of the triangle.



if anbody help me i would be very thankful to him






geometry analytic-geometry locus







share|cite|improve this question
























  • So the plane is always tangent to the sphere?
    – Michael Hoppe
    Aug 9 '17 at 5:53
















0












0








0








A plane moves so that its distance from the origin is a constant $p$. Write down the equation of the locus of the centroid of the triangle formed by its intersection with the three coordinates plane.




I was thinking about this question many times, but i could not get any hint to find the locus of the centroid of the triangle .
But i know that how to find the centroid of triangle we have to add (x+y+z/3,X+Y+Z/3), i don't know the locus of the centroid of the triangle.



if anbody help me i would be very thankful to him






geometry analytic-geometry locus







share|cite|improve this question
















A plane moves so that its distance from the origin is a constant $p$. Write down the equation of the locus of the centroid of the triangle formed by its intersection with the three coordinates plane.




I was thinking about this question many times, but i could not get any hint to find the locus of the centroid of the triangle .
But i know that how to find the centroid of triangle we have to add (x+y+z/3,X+Y+Z/3), i don't know the locus of the centroid of the triangle.



if anbody help me i would be very thankful to him






geometry analytic-geometry locus




geometry analytic-geometry locus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 9 '17 at 6:51









Michael Rozenberg

97.4k1589188




97.4k1589188










asked Aug 9 '17 at 5:45









lomberlomber

775320




775320












  • So the plane is always tangent to the sphere?
    – Michael Hoppe
    Aug 9 '17 at 5:53




















  • So the plane is always tangent to the sphere?
    – Michael Hoppe
    Aug 9 '17 at 5:53


















So the plane is always tangent to the sphere?
– Michael Hoppe
Aug 9 '17 at 5:53






So the plane is always tangent to the sphere?
– Michael Hoppe
Aug 9 '17 at 5:53












3 Answers
3






active

oldest

votes


















1














The equation of a variable plane in intercept form is $$frac xa+frac yb+ frac zc=1$$



Here, $a,b$ and $c$ are intercept on $x,y$ and $z$ axes.



Given, it's distance from origin is constant $$frac{Big|Big(frac xa+frac yb+ frac zc-1Big)_{x=y=z=0}Big|}{sqrt{frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}}}=p(text{constant})$$



And centroid of triangle $$(x,y,z)=left(frac a3,frac b3,frac c3right)$$



Therefore you get $$frac{1}{p^2}=frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}$$



Also, $a=3x,b=3y$ and $c=3z$



$$implies frac{1}{p^2}=frac {1}{9x^2}+frac {1}{9y^2}+frac {1}{9z^2}$$



Hence the desired locus is :
$$color{blue}{frac {1}{x^2}+frac {1}{y^2}+frac {1}{z^2}=frac{9}{p^2}}$$






share|cite|improve this answer























  • thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
    – lomber
    Aug 9 '17 at 5:57










  • What about the plane $z=p$?
    – Michael Hoppe
    Aug 9 '17 at 5:57










  • @MichaelHoppe I don't understand what you want to say.
    – Jaideep Khare
    Aug 9 '17 at 6:00










  • @lomberlego Check my edit.
    – Jaideep Khare
    Aug 9 '17 at 6:00










  • The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
    – Michael Hoppe
    Aug 9 '17 at 6:01



















0














Let $ac+by+cz+d=0$ be an equation of the plain.



Thus, $abcneq0$ and $$frac{|d|}{sqrt{a^2+b^2+c^2}}=p,$$
which gives $$|d|=psqrt{a^2+b^2+c^2}.$$



Let $Aleft(frac{-d}{a},0,0right)$, $Bleft(0,-frac{d}{b},0right)$ and $Cleft(0,0,-frac{d}{c}right)$ and for the centroid $M$ we obtain
$$Mleft(frac{-d}{3a},frac{-d}{3b},frac{-d}{3c}right),$$
which gives
$$frac{1}{x^2}+frac{1}{y^2}+frac{1}{z^2}=frac{9(a^2+b^2+c^2)}{d^2}=frac{9}{p^2}$$






share|cite|improve this answer























  • thanks a lots @ michael Rozenberg
    – lomber
    Aug 9 '17 at 13:18










  • @lomber lego You are welcome!
    – Michael Rozenberg
    Aug 9 '17 at 14:26



















0














Let $vec s$ be a vector of length $p$. Then the plane's equation is $langlevec s,vec xrangle=p^2$. Provided that no coordinate of $vec s$ is zero, the intercepts are obviously
$$frac{p^2}{s_k}vec e_kquadtext{for $k=1,2,3$}$$
where $vec e_k$ denote the $k^text{th}$ unit vector.



The result may be easily generalized to arbitrary dimensions.






share|cite|improve this answer





















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    3 Answers
    3






    active

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    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1














    The equation of a variable plane in intercept form is $$frac xa+frac yb+ frac zc=1$$



    Here, $a,b$ and $c$ are intercept on $x,y$ and $z$ axes.



    Given, it's distance from origin is constant $$frac{Big|Big(frac xa+frac yb+ frac zc-1Big)_{x=y=z=0}Big|}{sqrt{frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}}}=p(text{constant})$$



    And centroid of triangle $$(x,y,z)=left(frac a3,frac b3,frac c3right)$$



    Therefore you get $$frac{1}{p^2}=frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}$$



    Also, $a=3x,b=3y$ and $c=3z$



    $$implies frac{1}{p^2}=frac {1}{9x^2}+frac {1}{9y^2}+frac {1}{9z^2}$$



    Hence the desired locus is :
    $$color{blue}{frac {1}{x^2}+frac {1}{y^2}+frac {1}{z^2}=frac{9}{p^2}}$$






    share|cite|improve this answer























    • thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
      – lomber
      Aug 9 '17 at 5:57










    • What about the plane $z=p$?
      – Michael Hoppe
      Aug 9 '17 at 5:57










    • @MichaelHoppe I don't understand what you want to say.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • @lomberlego Check my edit.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
      – Michael Hoppe
      Aug 9 '17 at 6:01
















    1














    The equation of a variable plane in intercept form is $$frac xa+frac yb+ frac zc=1$$



    Here, $a,b$ and $c$ are intercept on $x,y$ and $z$ axes.



    Given, it's distance from origin is constant $$frac{Big|Big(frac xa+frac yb+ frac zc-1Big)_{x=y=z=0}Big|}{sqrt{frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}}}=p(text{constant})$$



    And centroid of triangle $$(x,y,z)=left(frac a3,frac b3,frac c3right)$$



    Therefore you get $$frac{1}{p^2}=frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}$$



    Also, $a=3x,b=3y$ and $c=3z$



    $$implies frac{1}{p^2}=frac {1}{9x^2}+frac {1}{9y^2}+frac {1}{9z^2}$$



    Hence the desired locus is :
    $$color{blue}{frac {1}{x^2}+frac {1}{y^2}+frac {1}{z^2}=frac{9}{p^2}}$$






    share|cite|improve this answer























    • thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
      – lomber
      Aug 9 '17 at 5:57










    • What about the plane $z=p$?
      – Michael Hoppe
      Aug 9 '17 at 5:57










    • @MichaelHoppe I don't understand what you want to say.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • @lomberlego Check my edit.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
      – Michael Hoppe
      Aug 9 '17 at 6:01














    1












    1








    1






    The equation of a variable plane in intercept form is $$frac xa+frac yb+ frac zc=1$$



    Here, $a,b$ and $c$ are intercept on $x,y$ and $z$ axes.



    Given, it's distance from origin is constant $$frac{Big|Big(frac xa+frac yb+ frac zc-1Big)_{x=y=z=0}Big|}{sqrt{frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}}}=p(text{constant})$$



    And centroid of triangle $$(x,y,z)=left(frac a3,frac b3,frac c3right)$$



    Therefore you get $$frac{1}{p^2}=frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}$$



    Also, $a=3x,b=3y$ and $c=3z$



    $$implies frac{1}{p^2}=frac {1}{9x^2}+frac {1}{9y^2}+frac {1}{9z^2}$$



    Hence the desired locus is :
    $$color{blue}{frac {1}{x^2}+frac {1}{y^2}+frac {1}{z^2}=frac{9}{p^2}}$$






    share|cite|improve this answer














    The equation of a variable plane in intercept form is $$frac xa+frac yb+ frac zc=1$$



    Here, $a,b$ and $c$ are intercept on $x,y$ and $z$ axes.



    Given, it's distance from origin is constant $$frac{Big|Big(frac xa+frac yb+ frac zc-1Big)_{x=y=z=0}Big|}{sqrt{frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}}}=p(text{constant})$$



    And centroid of triangle $$(x,y,z)=left(frac a3,frac b3,frac c3right)$$



    Therefore you get $$frac{1}{p^2}=frac {1}{a^2}+frac {1}{b^2}+frac {1}{c^2}$$



    Also, $a=3x,b=3y$ and $c=3z$



    $$implies frac{1}{p^2}=frac {1}{9x^2}+frac {1}{9y^2}+frac {1}{9z^2}$$



    Hence the desired locus is :
    $$color{blue}{frac {1}{x^2}+frac {1}{y^2}+frac {1}{z^2}=frac{9}{p^2}}$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Aug 9 '17 at 5:58

























    answered Aug 9 '17 at 5:53









    Jaideep KhareJaideep Khare

    17.7k32568




    17.7k32568












    • thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
      – lomber
      Aug 9 '17 at 5:57










    • What about the plane $z=p$?
      – Michael Hoppe
      Aug 9 '17 at 5:57










    • @MichaelHoppe I don't understand what you want to say.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • @lomberlego Check my edit.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
      – Michael Hoppe
      Aug 9 '17 at 6:01


















    • thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
      – lomber
      Aug 9 '17 at 5:57










    • What about the plane $z=p$?
      – Michael Hoppe
      Aug 9 '17 at 5:57










    • @MichaelHoppe I don't understand what you want to say.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • @lomberlego Check my edit.
      – Jaideep Khare
      Aug 9 '17 at 6:00










    • The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
      – Michael Hoppe
      Aug 9 '17 at 6:01
















    thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
    – lomber
    Aug 9 '17 at 5:57




    thanks a lot@ jaideep khare,, but i don't know where i haveto put this centroid coordinates and make a equation
    – lomber
    Aug 9 '17 at 5:57












    What about the plane $z=p$?
    – Michael Hoppe
    Aug 9 '17 at 5:57




    What about the plane $z=p$?
    – Michael Hoppe
    Aug 9 '17 at 5:57












    @MichaelHoppe I don't understand what you want to say.
    – Jaideep Khare
    Aug 9 '17 at 6:00




    @MichaelHoppe I don't understand what you want to say.
    – Jaideep Khare
    Aug 9 '17 at 6:00












    @lomberlego Check my edit.
    – Jaideep Khare
    Aug 9 '17 at 6:00




    @lomberlego Check my edit.
    – Jaideep Khare
    Aug 9 '17 at 6:00












    The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
    – Michael Hoppe
    Aug 9 '17 at 6:01




    The plane given by $z=p$ doesn't have an $x$- or a $y$-intercept.
    – Michael Hoppe
    Aug 9 '17 at 6:01











    0














    Let $ac+by+cz+d=0$ be an equation of the plain.



    Thus, $abcneq0$ and $$frac{|d|}{sqrt{a^2+b^2+c^2}}=p,$$
    which gives $$|d|=psqrt{a^2+b^2+c^2}.$$



    Let $Aleft(frac{-d}{a},0,0right)$, $Bleft(0,-frac{d}{b},0right)$ and $Cleft(0,0,-frac{d}{c}right)$ and for the centroid $M$ we obtain
    $$Mleft(frac{-d}{3a},frac{-d}{3b},frac{-d}{3c}right),$$
    which gives
    $$frac{1}{x^2}+frac{1}{y^2}+frac{1}{z^2}=frac{9(a^2+b^2+c^2)}{d^2}=frac{9}{p^2}$$






    share|cite|improve this answer























    • thanks a lots @ michael Rozenberg
      – lomber
      Aug 9 '17 at 13:18










    • @lomber lego You are welcome!
      – Michael Rozenberg
      Aug 9 '17 at 14:26
















    0














    Let $ac+by+cz+d=0$ be an equation of the plain.



    Thus, $abcneq0$ and $$frac{|d|}{sqrt{a^2+b^2+c^2}}=p,$$
    which gives $$|d|=psqrt{a^2+b^2+c^2}.$$



    Let $Aleft(frac{-d}{a},0,0right)$, $Bleft(0,-frac{d}{b},0right)$ and $Cleft(0,0,-frac{d}{c}right)$ and for the centroid $M$ we obtain
    $$Mleft(frac{-d}{3a},frac{-d}{3b},frac{-d}{3c}right),$$
    which gives
    $$frac{1}{x^2}+frac{1}{y^2}+frac{1}{z^2}=frac{9(a^2+b^2+c^2)}{d^2}=frac{9}{p^2}$$






    share|cite|improve this answer























    • thanks a lots @ michael Rozenberg
      – lomber
      Aug 9 '17 at 13:18










    • @lomber lego You are welcome!
      – Michael Rozenberg
      Aug 9 '17 at 14:26














    0












    0








    0






    Let $ac+by+cz+d=0$ be an equation of the plain.



    Thus, $abcneq0$ and $$frac{|d|}{sqrt{a^2+b^2+c^2}}=p,$$
    which gives $$|d|=psqrt{a^2+b^2+c^2}.$$



    Let $Aleft(frac{-d}{a},0,0right)$, $Bleft(0,-frac{d}{b},0right)$ and $Cleft(0,0,-frac{d}{c}right)$ and for the centroid $M$ we obtain
    $$Mleft(frac{-d}{3a},frac{-d}{3b},frac{-d}{3c}right),$$
    which gives
    $$frac{1}{x^2}+frac{1}{y^2}+frac{1}{z^2}=frac{9(a^2+b^2+c^2)}{d^2}=frac{9}{p^2}$$






    share|cite|improve this answer














    Let $ac+by+cz+d=0$ be an equation of the plain.



    Thus, $abcneq0$ and $$frac{|d|}{sqrt{a^2+b^2+c^2}}=p,$$
    which gives $$|d|=psqrt{a^2+b^2+c^2}.$$



    Let $Aleft(frac{-d}{a},0,0right)$, $Bleft(0,-frac{d}{b},0right)$ and $Cleft(0,0,-frac{d}{c}right)$ and for the centroid $M$ we obtain
    $$Mleft(frac{-d}{3a},frac{-d}{3b},frac{-d}{3c}right),$$
    which gives
    $$frac{1}{x^2}+frac{1}{y^2}+frac{1}{z^2}=frac{9(a^2+b^2+c^2)}{d^2}=frac{9}{p^2}$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Aug 9 '17 at 6:12

























    answered Aug 9 '17 at 6:04









    Michael RozenbergMichael Rozenberg

    97.4k1589188




    97.4k1589188












    • thanks a lots @ michael Rozenberg
      – lomber
      Aug 9 '17 at 13:18










    • @lomber lego You are welcome!
      – Michael Rozenberg
      Aug 9 '17 at 14:26


















    • thanks a lots @ michael Rozenberg
      – lomber
      Aug 9 '17 at 13:18










    • @lomber lego You are welcome!
      – Michael Rozenberg
      Aug 9 '17 at 14:26
















    thanks a lots @ michael Rozenberg
    – lomber
    Aug 9 '17 at 13:18




    thanks a lots @ michael Rozenberg
    – lomber
    Aug 9 '17 at 13:18












    @lomber lego You are welcome!
    – Michael Rozenberg
    Aug 9 '17 at 14:26




    @lomber lego You are welcome!
    – Michael Rozenberg
    Aug 9 '17 at 14:26











    0














    Let $vec s$ be a vector of length $p$. Then the plane's equation is $langlevec s,vec xrangle=p^2$. Provided that no coordinate of $vec s$ is zero, the intercepts are obviously
    $$frac{p^2}{s_k}vec e_kquadtext{for $k=1,2,3$}$$
    where $vec e_k$ denote the $k^text{th}$ unit vector.



    The result may be easily generalized to arbitrary dimensions.






    share|cite|improve this answer


























      0














      Let $vec s$ be a vector of length $p$. Then the plane's equation is $langlevec s,vec xrangle=p^2$. Provided that no coordinate of $vec s$ is zero, the intercepts are obviously
      $$frac{p^2}{s_k}vec e_kquadtext{for $k=1,2,3$}$$
      where $vec e_k$ denote the $k^text{th}$ unit vector.



      The result may be easily generalized to arbitrary dimensions.






      share|cite|improve this answer
























        0












        0








        0






        Let $vec s$ be a vector of length $p$. Then the plane's equation is $langlevec s,vec xrangle=p^2$. Provided that no coordinate of $vec s$ is zero, the intercepts are obviously
        $$frac{p^2}{s_k}vec e_kquadtext{for $k=1,2,3$}$$
        where $vec e_k$ denote the $k^text{th}$ unit vector.



        The result may be easily generalized to arbitrary dimensions.






        share|cite|improve this answer












        Let $vec s$ be a vector of length $p$. Then the plane's equation is $langlevec s,vec xrangle=p^2$. Provided that no coordinate of $vec s$ is zero, the intercepts are obviously
        $$frac{p^2}{s_k}vec e_kquadtext{for $k=1,2,3$}$$
        where $vec e_k$ denote the $k^text{th}$ unit vector.



        The result may be easily generalized to arbitrary dimensions.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 9 '17 at 6:19









        Michael HoppeMichael Hoppe

        10.8k31834




        10.8k31834






























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            Mont Emei

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            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
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            Province de Neuquén

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            Province de Neuquén Provincia del Neuquén .mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Héraldique Drapeau Localisation de la province de Neuquén Administration Pays   Argentine Capitale Neuquén Gouverneur Omar Gutiérrez ISO 3166-2 AR-Q Démographie Gentilé Neuquino/a Population 550 344  hab. (2010) Densité 5,8  hab./km 2 Géographie Superficie 94 078  km 2 Liens Site web http://www.neuquen.gov.ar modifier   La province de Neuquén est une province d'Argentine, qui se trouve à l'extrémité nord-ouest de la Patagonie argentine. Avec une superficie de 94 378  km 2 , elle est la plus petite province des régions australes du pays. Sa géographie s'abîme sur l'imposante Cordillère des Andes. Sommaire 1 Histoire 1.1 Avant la découverte par les Espagnols 1.2 ...
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            Journaliste

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            Pour les articles homonymes, voir Journaliste (homonymie). Journaliste Journalistes Appellation Journaliste Secteur d'activité Média - Communication Professions voisines Journaliste sportif - Reporter-photographe - Éditorialiste Codes CITP 2451, 2642 IDEO (France) 10197 ROME (France) E1106 modifier - modifier le code - modifier Wikidata Un journaliste est une personne qui recueille, écrit ou distribue des informations. L'activité d'un journaliste est le journalisme. Un journaliste peut travailler sur des sujets généraux ou se spécialiser dans certaines questions. La plupart des journalistes ont tendance à se spécialiser et, en collaborant avec d'autres journalistes, rédigent un article ou mettent en forme un reportage qui couvre l'actualité. Par exemple, un journaliste sportif couvre les nouvelles dans le monde du sport, mais ce journaliste peut aussi faire partie d'un journal qui couvre d'autres su...
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