MHB Triangle with Complex Number Vertices SATISFIES Equilateral Relation

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The discussion centers on a problem of proving that a triangle with vertices represented by complex numbers \(a\), \(b\), and \(c\) is equilateral if it satisfies the equation \(\dfrac{a-b}{c-b}+\dfrac{c-a}{b-a}=2\left(\dfrac{b-c}{a-c}\right)\). Participants engage in solving this problem, with Opalg providing a correct solution. The focus is on the mathematical proof and the implications of the given equation. The thread highlights the relationship between complex numbers and geometric properties of triangles. Ultimately, the proof confirms that the specified condition indeed leads to an equilateral triangle.
anemone
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Here is this week's POTW:

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If the vertices of a triangle are represented by the complex numbers $a,\,b,\,c$ and these numbers satisfy $\dfrac{a-b}{c-b}+\dfrac{c-a}{b-a}=2\left(\dfrac{b-c}{a-c}\right)$, prove that the triangle is equilateral.

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Congratulations to Opalg for his correct solution(Cool), which you can find below:
Let the sides of the triangle be $x = b-c$, $y = c-a$ and $z = a-b$. Then $x+y+z = 0$, and the given equation becomes $\dfrac zx + \dfrac yz = 2\dfrac xy$. Multiply by $xyz$ to get $$z^2y + y^2x = 2x^2z.$$ Now substitute $z = -(x+y)$: $$ (x+y)^2y + y^2x + 2x^2(x+y) = 0, $$ $$ y^3 + 3y^2x + 3yx^2 + 2x^3 = 0.$$ Divide by $x^3$ and let $\lambda = \dfrac yx$: $$\lambda^3 + 3\lambda^2 + 3\lambda + 2 = 0,$$ $$(\lambda+1)^3 = -1.$$ Let $\omega = e^{2\pi i/3}$. The complex cube roots of $1$ are $\omega^k \ (k = 0,1,2)$. Take the cube roots of the previous equation to see that the three values of $\lambda + 1$ are $-1$, $-\omega$ and $-\omega^2$. Therefore $\lambda = -2$, $\lambda = -1 - \omega = \omega^2$ or $\lambda = -1-\omega^2 = \omega$. If $\lambda=-2$ then $y=-2x$ and $z=x$ (because $x+y+z=0$). So $x$, $y$ and $z$ lie on a straight line and cannot be the sides of a triangle. If $\lambda=\omega$ then $y=\omega x$ and $z=\omega^2x$ (because $1 + \omega + \omega^2=0$). Similarly, if $\lambda=\omega^2$ then $y=\omega^2 x$ and $z=\omega x$. In both cases, $x$, $y$ and $z$ all have the same absolute value, and their arguments differ by $120^\circ$. So they form the sides of an equilateral triangle (with vertices at $a$, $b$ and $c$).
 
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