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Reply to "why is PI a constant?"
Posted:
Jun 12, 2001 12:27 PM
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Let (opqr) be a square of side length L,such that op=pq=qr=ro=L.
Diagonals (oq) and (pr) cross at g.Inscribed circle touches square
at mid point of ( op) at a,(pq )at b, (qr) at c, and (ro) at d.
Choose a point F between (oa).
Assume, that ,gF is the radius of the circle ,that,has a circumference
equal to that of the perimeter of the square (4L).
This circle crosses ,also, the square at point:
G between a and p,
H " p " b,
I " b " q,
J " q " c,
K " c " r,
L " r " d,
M " d " o.
Draw FgJ=D= circle's diameter,and FK=square side=L. Call KJ=E.
Orthogonal triangle FKJ, has angle FJK ,called THETA.
D= sqrt[L^2+E^2],
[D/L]=sqrt[1+(E^2/L^2)] , call [L/E]=tan(THETA)
or,
(4L/D)=Pi=4/{sqrt[1+(E^2/L^2)]}.
We see, thus , that Pi, is a constant.
[For the current value of Pi=3.141592654.. , the value of THETA
equals to : 51.75751852..degrees.]
This is my own proof that,Pi,is a CONSTANT.
As it is obvious the solution touches ,the linear part,of quadrature.
Regards,
Panagiotis Stefanides
http://www.stefanides.gr
http://www.dotcreative.com/stefanides/quad.htm
Copyright Panagiotis Stefanides 2001.
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