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From: moroney@world.std.spaamtrap.com (Michael Moroney)
Newsgroups: sci.math,sci.physics
Subject: Re: 8-teaching Logic in Grade School, TEACHING TRUE MATHEMATICS
Date: Fri, 25 May 2018 20:29:57 +0000 (UTC)
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Archimedes Plutonium <plutonium.archimedes@gmail.com> writes:


>#3-9  Many examples and exercises in using Algebraic Logic // Teaching 
>True Mathematics

>These last two lessons are going to be long lessons

Here is some True Mathematics which will probably become long lessons:

Below you will find a simple *proof* that shows that certain conic 
sections are ellipses.

Some preliminaries:

Top view of the conic section and depiction of the coordinate system used 
in the proof:

              ^ x
              |
             -+- <= x=h
         .'   |   `.
        .     |     .
        |     |     |
        '     |     '      
         `.   |   .'
 y <----------+ <= x=0
             
Cone (side view):
                 .
                /|\
               / | \
              /b |  \
             /---+---' <= x = h
            /    |'   \
           /   ' |     \
          / '    |      \
x = 0 => '-------+-------\
        /    a   |        \

Proof:

r(x) = a - ((a-b)/h)x  and  d(x) = a - ((a+b)/h)x,  hence

y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h/2)^2/h^2.

Hence (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1  ...equation of an ellipse

qed