document.write( "Question 1120925: The roots z1,z2 Nd z3 of the equation x^3+3ax^2+3bx+c=0 in which a,b and c are complex numbers, correspond to the points A,B and C on the Argand plane. Find the C.G. of ∆ABC and show that it will be equilateral if a^2=b \n" ); document.write( "
Algebra.Com's Answer #736630 by ikleyn(52788)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Find the C.G. of ∆ABC\r
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document.write( "C.G. of a triangle is the center of gravity of the triangle.\r\n" );
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document.write( "The center of gravity of the triangle with the vertices A, B and C is the point with coordinates\r\n" );
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document.write( "    \"%281%2F3%29%2A%28x%5BA%5D+%2B+x%5BB%5D+%2B+x%5BC%5D%29\"   and   \"%281%2F3%29%2A%28y%5BA%5D+%2B+y%5BB%5D+%2B+y%5BC%5D%29\" .\r\n" );
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document.write( "Geometrically, this point is nothing else as   \"%281%2F3%29%2A%28z%5B1%5D+%2B+z%5B2%5D+%2B+z%5B3%5D%29\".\r\n" );
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document.write( "According to Vieta's theorem,  the sum  \"%28z%5B1%5D+%2B+z%5B2%5D+%2B+z%5B3%5D%29\"  is the coefficient at  x^2  of the given equation  with the opposite sign, i.e.  -3a.\r\n" );
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document.write( "Thus the center of gravity of the given triangle is the complex number  -a.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Show that it will be equilateral if a^2 = b\r
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document.write( "If  a^2 = b,  then  the given equation\r\n" );
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document.write( "    x^3 + 3ax^2 + 3bx + c = 0\r\n" );
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document.write( "becomes\r\n" );
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document.write( "    x^3 + 3ax^2 + 3a^2*x + c = 0,    or, equivalently,\r\n" );
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document.write( "    (x^3 + 3ax^3 + 3a^2*x + a^3) + (c-a^3) = 0,   \r\n" );
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document.write( "    (x+a)^3 = -(c-a^3),\r\n" );
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document.write( "    x + a = \"root%283%2Ca%5E3-c%29\",\r\n" );
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document.write( "    x = - a + \"root%283%2Ca%5E3-c%29\".     (1)\r\n" );
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document.write( "If you are familiar with the basics of complex number theory, you will recognize that the formula (1)\r\n" );
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document.write( "describes the three complex numbers around the point \"-a\"  (the center of gravity of the triangle ABC) turned at the angle 120 degrees\r\n" );
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document.write( "one relative the other.\r\n" );
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document.write( "So the triangle ABC is an equilateral triangle.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding the basics of complex number theory,  you have the set of lessons\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
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\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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