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MathGroup Archive 2006

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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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