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Transformstion to canonical form
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65431] Transformstion to canonical form
*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>
*Date*: Fri, 31 Mar 2006 06:09:44 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
The following rather simple problem led my to some genral questions of
how to proceed in symbolic transformations which I would do on paper
within Mathematica. I'm sure you can give useful hints which I apreciate
in advance.
Consider a slight generalization of the Thales circle: In a triangle
ABC, let g be the fixed angle in vertex C. (In the Thales case we have g
= Pi/2).
What is the equation for the coordinates (x,y) of the point C?
The immediate statement is derived from the scalar product of the two
vectors (C-A) and (C-B):
(1)
(C-A).(C-B)== Sqrt[(C-A).C-A)] Sqrt[(C-B).(C-A)]
where
(2)
G = Cos[g]
Putting
A={0,0}; B={b,0};C={x,y} we obtain the equation:
eq1 = x^2 - b*x + y^2 == G*Sqrt[x^2 + y^2]*Sqrt[(x - b)^2 + y^2]
Now: what is a useful procedure in Mathematica to transform eq1 into an
equation of the form
eq2 = (x - u)^2 + (y - v)^2 == w
and to find the parameters u, v, and w in terms of b and G?
Regards,
Wolfgang
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