Proof that the triangle formed by mass centers is equilateral. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Centroids of triangleCompetition style problem circa 1992there concurrent lines, perpendicular to the sides of a triangleProve that $S = 8S_1-4S_2$ in the geometrical figureCircle inscribed in Equilateral TrianglesProve triangle similiarity by given expressionAAA similarity Theorem.Similarity of nested trianglesProving similarity for a triangle inside another triangle given ratio between sides.Triangle equilateral proof

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Proof that the triangle formed by mass centers is equilateral.



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Centroids of triangleCompetition style problem circa 1992there concurrent lines, perpendicular to the sides of a triangleProve that $S = 8S_1-4S_2$ in the geometrical figureCircle inscribed in Equilateral TrianglesProve triangle similiarity by given expressionAAA similarity Theorem.Similarity of nested trianglesProving similarity for a triangle inside another triangle given ratio between sides.Triangle equilateral proof










2












$begingroup$


Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals.
Consider now $A_2$, $B_2$ and $C_2$ mass centres of $BCA_1$, $ACB_1$ and $ABC1$, respectively. Prove that triangle $A_2B_2C_2$ is a equilateral triangle.










share|cite|improve this question











$endgroup$
















    2












    $begingroup$


    Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals.
    Consider now $A_2$, $B_2$ and $C_2$ mass centres of $BCA_1$, $ACB_1$ and $ABC1$, respectively. Prove that triangle $A_2B_2C_2$ is a equilateral triangle.










    share|cite|improve this question











    $endgroup$














      2












      2








      2





      $begingroup$


      Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals.
      Consider now $A_2$, $B_2$ and $C_2$ mass centres of $BCA_1$, $ACB_1$ and $ABC1$, respectively. Prove that triangle $A_2B_2C_2$ is a equilateral triangle.










      share|cite|improve this question











      $endgroup$




      Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals.
      Consider now $A_2$, $B_2$ and $C_2$ mass centres of $BCA_1$, $ACB_1$ and $ABC1$, respectively. Prove that triangle $A_2B_2C_2$ is a equilateral triangle.







      geometry euclidean-geometry centroid geometric-transformation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Apr 2 at 19:02









      SMM

      3,268512




      3,268512










      asked Apr 2 at 18:03









      Andrei AlexandruAndrei Alexandru

      134




      134




















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          Consider the compositions of rotations $I= R_A_2,120^circcirc R_B_2,120^circ$; note $I(A)=B$. Let $S$ be the point such that oriented angle $angle(SB_2,B_2A_2)=60^circ$ and oriented angle $angle (B_2A_2,SA_2)=60^circ$. We have:
          $$I=R_A_2,120^circcirc R_B_2,120^circ= S_SA_2circ S_A_2B_2circ S_A_2B_2circ S_SB_2= S_SA_2circ S_SB_2= R_S,240^circ.$$
          Therefore, $R_S,240^circ(A)=B$, so $SA=SB$ and oriented angle $angle(SA,SB)=240^circ$. Now we conclude that $S=C_2$. So $angle(C_2B_2,B_2A_2)=60^circ$ and $angle (B_2A_2,C_2A_2)=60^circ$, hence $triangle A_2B_2C_2$ is equilateral.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1...............
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:29


















          1












          $begingroup$

          I am working with $A,B,C$ are oriented clockwise, so you can make a sketch. We have $A_2=(B+C+A_1)/3$, $B_2=(A+B+C_1)/3$, $C_2=(A+C+B_1)/3$.



          We want to show that if we rotate $overlineB_2C_2$ an angle of 60 anti-clockwise we get $overlineB_2A_2$. In other words want to show that $R(C_2-B_2)=A_2-B_2$, where $R$ denotes the operation rotation by 60 degrees(wrt origin), note this is a linear map .
          Let's compute beginalignR(C_2-B_2)&=R(fracC+B_1-B-C_13)=R(fracC-B+B_1-A+A-C_13)\
          &=fracR(C-B)3+fracR(B_1-A)3+fracR(A-C_1)3
          endalign

          Now, note that $R(C-B)=A_1-B$, $R(B_1-A)=C-A$ and $R(A-C_1)=B-C_1$ because $BCA_1$, $ACB_1$ and $ABC_1$ are equilateral.
          We get
          beginalignR(C_2-B_2)=fracA_1-B3+fracC-A3+fracB-C_13=fracB+C+A_13-fracA+B+C_13=A_2-B_2.
          endalign

          This is what we wanted to prove.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1..................
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a 5th grade proof using only elementary geometry
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:37










          • $begingroup$
            This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
            $endgroup$
            – Julian Mejia
            Apr 2 at 19:44












          Your Answer








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          2 Answers
          2






          active

          oldest

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          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          Consider the compositions of rotations $I= R_A_2,120^circcirc R_B_2,120^circ$; note $I(A)=B$. Let $S$ be the point such that oriented angle $angle(SB_2,B_2A_2)=60^circ$ and oriented angle $angle (B_2A_2,SA_2)=60^circ$. We have:
          $$I=R_A_2,120^circcirc R_B_2,120^circ= S_SA_2circ S_A_2B_2circ S_A_2B_2circ S_SB_2= S_SA_2circ S_SB_2= R_S,240^circ.$$
          Therefore, $R_S,240^circ(A)=B$, so $SA=SB$ and oriented angle $angle(SA,SB)=240^circ$. Now we conclude that $S=C_2$. So $angle(C_2B_2,B_2A_2)=60^circ$ and $angle (B_2A_2,C_2A_2)=60^circ$, hence $triangle A_2B_2C_2$ is equilateral.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1...............
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:29















          1












          $begingroup$

          Consider the compositions of rotations $I= R_A_2,120^circcirc R_B_2,120^circ$; note $I(A)=B$. Let $S$ be the point such that oriented angle $angle(SB_2,B_2A_2)=60^circ$ and oriented angle $angle (B_2A_2,SA_2)=60^circ$. We have:
          $$I=R_A_2,120^circcirc R_B_2,120^circ= S_SA_2circ S_A_2B_2circ S_A_2B_2circ S_SB_2= S_SA_2circ S_SB_2= R_S,240^circ.$$
          Therefore, $R_S,240^circ(A)=B$, so $SA=SB$ and oriented angle $angle(SA,SB)=240^circ$. Now we conclude that $S=C_2$. So $angle(C_2B_2,B_2A_2)=60^circ$ and $angle (B_2A_2,C_2A_2)=60^circ$, hence $triangle A_2B_2C_2$ is equilateral.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1...............
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:29













          1












          1








          1





          $begingroup$

          Consider the compositions of rotations $I= R_A_2,120^circcirc R_B_2,120^circ$; note $I(A)=B$. Let $S$ be the point such that oriented angle $angle(SB_2,B_2A_2)=60^circ$ and oriented angle $angle (B_2A_2,SA_2)=60^circ$. We have:
          $$I=R_A_2,120^circcirc R_B_2,120^circ= S_SA_2circ S_A_2B_2circ S_A_2B_2circ S_SB_2= S_SA_2circ S_SB_2= R_S,240^circ.$$
          Therefore, $R_S,240^circ(A)=B$, so $SA=SB$ and oriented angle $angle(SA,SB)=240^circ$. Now we conclude that $S=C_2$. So $angle(C_2B_2,B_2A_2)=60^circ$ and $angle (B_2A_2,C_2A_2)=60^circ$, hence $triangle A_2B_2C_2$ is equilateral.






          share|cite|improve this answer









          $endgroup$



          Consider the compositions of rotations $I= R_A_2,120^circcirc R_B_2,120^circ$; note $I(A)=B$. Let $S$ be the point such that oriented angle $angle(SB_2,B_2A_2)=60^circ$ and oriented angle $angle (B_2A_2,SA_2)=60^circ$. We have:
          $$I=R_A_2,120^circcirc R_B_2,120^circ= S_SA_2circ S_A_2B_2circ S_A_2B_2circ S_SB_2= S_SA_2circ S_SB_2= R_S,240^circ.$$
          Therefore, $R_S,240^circ(A)=B$, so $SA=SB$ and oriented angle $angle(SA,SB)=240^circ$. Now we conclude that $S=C_2$. So $angle(C_2B_2,B_2A_2)=60^circ$ and $angle (B_2A_2,C_2A_2)=60^circ$, hence $triangle A_2B_2C_2$ is equilateral.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Apr 2 at 18:59









          SMMSMM

          3,268512




          3,268512











          • $begingroup$
            +1...............
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:29
















          • $begingroup$
            +1...............
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:29















          $begingroup$
          +1...............
          $endgroup$
          – Maria Mazur
          Apr 2 at 19:02




          $begingroup$
          +1...............
          $endgroup$
          – Maria Mazur
          Apr 2 at 19:02












          $begingroup$
          I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
          $endgroup$
          – Andrei Alexandru
          Apr 2 at 19:29




          $begingroup$
          I need a proof for 5th grade. You can't use rotations or this kind of abstract tools
          $endgroup$
          – Andrei Alexandru
          Apr 2 at 19:29











          1












          $begingroup$

          I am working with $A,B,C$ are oriented clockwise, so you can make a sketch. We have $A_2=(B+C+A_1)/3$, $B_2=(A+B+C_1)/3$, $C_2=(A+C+B_1)/3$.



          We want to show that if we rotate $overlineB_2C_2$ an angle of 60 anti-clockwise we get $overlineB_2A_2$. In other words want to show that $R(C_2-B_2)=A_2-B_2$, where $R$ denotes the operation rotation by 60 degrees(wrt origin), note this is a linear map .
          Let's compute beginalignR(C_2-B_2)&=R(fracC+B_1-B-C_13)=R(fracC-B+B_1-A+A-C_13)\
          &=fracR(C-B)3+fracR(B_1-A)3+fracR(A-C_1)3
          endalign

          Now, note that $R(C-B)=A_1-B$, $R(B_1-A)=C-A$ and $R(A-C_1)=B-C_1$ because $BCA_1$, $ACB_1$ and $ABC_1$ are equilateral.
          We get
          beginalignR(C_2-B_2)=fracA_1-B3+fracC-A3+fracB-C_13=fracB+C+A_13-fracA+B+C_13=A_2-B_2.
          endalign

          This is what we wanted to prove.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1..................
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a 5th grade proof using only elementary geometry
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:37










          • $begingroup$
            This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
            $endgroup$
            – Julian Mejia
            Apr 2 at 19:44
















          1












          $begingroup$

          I am working with $A,B,C$ are oriented clockwise, so you can make a sketch. We have $A_2=(B+C+A_1)/3$, $B_2=(A+B+C_1)/3$, $C_2=(A+C+B_1)/3$.



          We want to show that if we rotate $overlineB_2C_2$ an angle of 60 anti-clockwise we get $overlineB_2A_2$. In other words want to show that $R(C_2-B_2)=A_2-B_2$, where $R$ denotes the operation rotation by 60 degrees(wrt origin), note this is a linear map .
          Let's compute beginalignR(C_2-B_2)&=R(fracC+B_1-B-C_13)=R(fracC-B+B_1-A+A-C_13)\
          &=fracR(C-B)3+fracR(B_1-A)3+fracR(A-C_1)3
          endalign

          Now, note that $R(C-B)=A_1-B$, $R(B_1-A)=C-A$ and $R(A-C_1)=B-C_1$ because $BCA_1$, $ACB_1$ and $ABC_1$ are equilateral.
          We get
          beginalignR(C_2-B_2)=fracA_1-B3+fracC-A3+fracB-C_13=fracB+C+A_13-fracA+B+C_13=A_2-B_2.
          endalign

          This is what we wanted to prove.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            +1..................
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a 5th grade proof using only elementary geometry
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:37










          • $begingroup$
            This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
            $endgroup$
            – Julian Mejia
            Apr 2 at 19:44














          1












          1








          1





          $begingroup$

          I am working with $A,B,C$ are oriented clockwise, so you can make a sketch. We have $A_2=(B+C+A_1)/3$, $B_2=(A+B+C_1)/3$, $C_2=(A+C+B_1)/3$.



          We want to show that if we rotate $overlineB_2C_2$ an angle of 60 anti-clockwise we get $overlineB_2A_2$. In other words want to show that $R(C_2-B_2)=A_2-B_2$, where $R$ denotes the operation rotation by 60 degrees(wrt origin), note this is a linear map .
          Let's compute beginalignR(C_2-B_2)&=R(fracC+B_1-B-C_13)=R(fracC-B+B_1-A+A-C_13)\
          &=fracR(C-B)3+fracR(B_1-A)3+fracR(A-C_1)3
          endalign

          Now, note that $R(C-B)=A_1-B$, $R(B_1-A)=C-A$ and $R(A-C_1)=B-C_1$ because $BCA_1$, $ACB_1$ and $ABC_1$ are equilateral.
          We get
          beginalignR(C_2-B_2)=fracA_1-B3+fracC-A3+fracB-C_13=fracB+C+A_13-fracA+B+C_13=A_2-B_2.
          endalign

          This is what we wanted to prove.






          share|cite|improve this answer









          $endgroup$



          I am working with $A,B,C$ are oriented clockwise, so you can make a sketch. We have $A_2=(B+C+A_1)/3$, $B_2=(A+B+C_1)/3$, $C_2=(A+C+B_1)/3$.



          We want to show that if we rotate $overlineB_2C_2$ an angle of 60 anti-clockwise we get $overlineB_2A_2$. In other words want to show that $R(C_2-B_2)=A_2-B_2$, where $R$ denotes the operation rotation by 60 degrees(wrt origin), note this is a linear map .
          Let's compute beginalignR(C_2-B_2)&=R(fracC+B_1-B-C_13)=R(fracC-B+B_1-A+A-C_13)\
          &=fracR(C-B)3+fracR(B_1-A)3+fracR(A-C_1)3
          endalign

          Now, note that $R(C-B)=A_1-B$, $R(B_1-A)=C-A$ and $R(A-C_1)=B-C_1$ because $BCA_1$, $ACB_1$ and $ABC_1$ are equilateral.
          We get
          beginalignR(C_2-B_2)=fracA_1-B3+fracC-A3+fracB-C_13=fracB+C+A_13-fracA+B+C_13=A_2-B_2.
          endalign

          This is what we wanted to prove.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Apr 2 at 18:49









          Julian MejiaJulian Mejia

          76729




          76729











          • $begingroup$
            +1..................
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a 5th grade proof using only elementary geometry
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:37










          • $begingroup$
            This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
            $endgroup$
            – Julian Mejia
            Apr 2 at 19:44

















          • $begingroup$
            +1..................
            $endgroup$
            – Maria Mazur
            Apr 2 at 19:02










          • $begingroup$
            I need a 5th grade proof using only elementary geometry
            $endgroup$
            – Andrei Alexandru
            Apr 2 at 19:37










          • $begingroup$
            This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
            $endgroup$
            – Julian Mejia
            Apr 2 at 19:44
















          $begingroup$
          +1..................
          $endgroup$
          – Maria Mazur
          Apr 2 at 19:02




          $begingroup$
          +1..................
          $endgroup$
          – Maria Mazur
          Apr 2 at 19:02












          $begingroup$
          I need a 5th grade proof using only elementary geometry
          $endgroup$
          – Andrei Alexandru
          Apr 2 at 19:37




          $begingroup$
          I need a 5th grade proof using only elementary geometry
          $endgroup$
          – Andrei Alexandru
          Apr 2 at 19:37












          $begingroup$
          This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
          $endgroup$
          – Julian Mejia
          Apr 2 at 19:44





          $begingroup$
          This is called Napoleon's theorem, I bet you can find geometric proofs if you search for it. But the proof I gave is understandable for anyone as long as you admit that rotation of sum of vectors is a sum of the vectors rotated. Anyway, good luck.In fact, If you draw the vectors I use would be very enlightening.
          $endgroup$
          – Julian Mejia
          Apr 2 at 19:44


















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