Complex numbers and geometry
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04-22-2010, 09:50 PM
Post: #1
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Complex numbers and geometry
(1) Let \(a,b,c \in \mathbf{C}, \quad |a|=|b|=|c|=1\), show that \(a,b,c \) represent vertices of a equilateral triangle iff
\(\quad \quad \quad a+b+c=0\) (2) Find the geometry corresponding to (a) \(I(z^2)>2\); (b) \(\left|\displaystyle{\frac{z-1}{z+1}}\right|<1\) (3)A triangle's two vertices are \(a,b\), find the region for the 3rd vertex \(z\). (4) If $\sum_1^6 z_j = 0,\; |z_k| \le 1 \; (\forall k)$ then $|x_i+x_j+x_k|\le 1$ for some distinct $i,j,k \in \{1,2,3,4,5,6\}$ |
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04-23-2010, 09:31 AM
Post: #2
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RE: Complex numbers and geometry
Let $x,y,z$ be any permutation of $a,b,c$, we have
(1)"If" part: $\overline{x}(x+y+z)=0 \Rightarrow \overline{x}(y+z)=-1=x( \overline{y}+\overline{z})$ So $\quad |x-y|^{2}-|y-z|^{2}= ((x-y)(\overline{x}-\overline{y})+(x-z)(\overline{x}-\overline{z}))-((y-z)(\overline{y}-\overline{z})+(x-z)(\overline{x}-\overline{z}))$ $\quad =(4-x(\overline{y}+\overline{z})-\overline{x}(y+z))-(4-z(\overline{x}+\overline{y})-\overline{z}(x+y))=6-6=0$ $\quad $thus $\quad \quad |x-y|=|y-z|$ $\quad$If only: Let $\sigma = x+y+z, \quad \rho = |\sigma|$ $\quad |x-y|^2+|x-z|^2=|y-z|^2+|x-z|^2 \Rightarrow x( \overline{y}+\overline{z})+\overline{x}(y+z)=z( \overline{x}+\overline{y})+\overline{z}(x+y)$ $\quad \overset{2x \overline{x}=2z \overline{z}}{\Rightarrow} x \overline{\sigma}+\overline{x}\sigma=z\overline{\sigma}+\overline{z}\sigma \Rightarrow (x-z) \overline{\sigma} = -( \overline{x-z} ) \sigma \Rightarrow ((x-z) / |x-z|)^2 \rho^2 = \sigma^2$ $\quad \quad \Rightarrow \sigma = 0$ since \(x,y,z\) is an arbitrary permutation of \(a,b,c\) |
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