Question #f312d

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Question #f312d

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Answer 1 · 2017-04-21T17:24:37

$4/(1 - sqrt[5]) + 4/(1 + sqrt[5])=-2$

Explanation:

Given $z_1$ we can calculate $z_2$ according to

$4z_2^2+z_1^2-2iz_1z_2=0$ or

$z_2 =i/4(1pm sqrt(5)) z_1$

which means that given $z_1$ to obtain $z_2$ we need two transformations
1) Scaling by $1/4(1pm sqrt5)$
2) Rotating $pi/2$ counterclockwise as associated to the product by $i$

so with $z_0 = 0$ we can build two triangles

$[z_0,z_1,z_2^a]$ and $[z_0,z_1,z_2^b]$

with

$z_2^a=i/4(1+ sqrt(5)) z_1$
$z_2^b=i/4(1- sqrt(5)) z_1$

so $z_2^a,z_0,z_2^b$ are aligned and

$[z_2^a,z_2^b]$ is perpendicular to $[z_0,z_1]$

and given $(z_1)_k$ all triangles $[z_2^a,z_1,z_2^b]_k$ are similar.

The least angles at $[z_2^a,z_0,z_1]$ and $[z_2^b,z_0,z_1]$ are respectively

$alpha = arctan((1-sqrt(5))/4), beta = arctan((1+sqrt(5))/4)$

and

$cot(alpha)+cot(beta) = 4/(1 - sqrt[5]) + 4/(1 + sqrt[5])=-2$

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Question #f312d

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