Question 1120925
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<U>Find the C.G. of &#8710;ABC</U>


<pre>
C.G. of a triangle is the <U>center of gravity</U> of the triangle.


The center of gravity of the triangle with the vertices A, B and C is the point with coordinates


    {{{(1/3)*(x[A] + x[B] + x[C])}}}   and   {{{(1/3)*(y[A] + y[B] + y[C])}}} .


Geometrically, this point is nothing else as   {{{(1/3)*(z[1] + z[2] + z[3])}}}.


According to Vieta's theorem,  the sum  {{{(z[1] + z[2] + z[3])}}}  is the coefficient at  x^2  of the given equation  with the opposite sign, i.e.  -3a.


Thus the center of gravity of the given triangle is the complex number  -a.
</pre>


<U>Show that it will be equilateral if a^2 = b</U>


<pre>
If  a^2 = b,  then  the given equation


    x^3 + 3ax^2 + 3bx + c = 0


becomes


    x^3 + 3ax^2 + 3a^2*x + c = 0,    or, equivalently,


    (x^3 + 3ax^3 + 3a^2*x + a^3) + (c-a^3) = 0,   

    (x+a)^3 = -(c-a^3),

    x + a = {{{root(3,a^3-c)}}},


    x = - a + {{{root(3,a^3-c)}}}.     (1)


If you are familiar with the basics of complex number theory, you will recognize that the formula (1)

describes the three complex numbers around the point "-a"  (the center of gravity of the triangle ABC) turned at the angle 120 degrees

one relative the other.


So the triangle ABC is an equilateral triangle.
</pre>


Regarding the basics of complex number theory, &nbsp;you have the set of lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-real-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with real coefficients on complex domain</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson>How to take a square root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-complex-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with complex coefficients on complex domain</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Complex numbers</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.