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Alike Triangles
Posted:
Nov 29, 2010 10:15 AM


Consider the fucntion f(x)=2* sin^1(mod[x/2]).
We define ?Alike Triangle? for any root @ of any equation as isosceles triangle with vertical angle equal to f(@). For example equation x^3 ? 3*x + 1=0 has roots 2sin(pi/18), 2sin(5pi/18) and 2sin(7pi/18). So for root 2sin(7pi/18) corresponding alike triangle will be isosceles triangle with vertical angle equal to f(2sin(7pi/18))=2*sin^1(mod[{2sin(7pi/18)}/2])=2*sin^1(mod[sin(7pi/18)])=2*sin^1(sin(7pi/18)) =2*(7pi/18) In the same way, ?Alike Triangles? for equation x^3 ? 3*x + 1=0 will be isosceles triangle with vertical angles 20 degree, 100 degree and 140 degrre. We have seen these triangles have some alike properties.(See my post with title Puzzling Geometry). Here I am not giving exact definition of alike properties.
Q.Find alike triangles and their alike properties corresponding to following 2 equations x^2 x 1=0 and x^3 ? x^2 2*x +1=0.



