       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

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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