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My Brother Noel's Octagon Proof

 

My three year younger brother, Noel, after studying my unproven observations and make up of octagon formulas now proves that my perceptions and formulas do have a basis in fact. (BIF)

 

Since Noel as a young teenager designed and built functional radio receivers, long distance radio transmitters and radio directional antennas I've known that Noel was exceptionally gifted and talented in the fields of optics, electronics, mathematics and original thinking.

 

In Noel's own words, here is his proof of my, until now, unproven octagon formulas.

 

Noel Reekie's Proof that the Side of a Regular Octagon is Equal to 0.4142 Times the Width of the Octagon.

 

 

First prove that EF = BD

 

If AC and OC = 1

Then AO and AD = the square root of 2

Therefore CD = the square root of 2 minus 1

Similarly BC = the square root of 2 minus 1

Since AE = 2

Then EF = 2 times the square root of 2 minus 1

As does BD = 2 times the square root of 2 minus 1

Therefore EF = BD

 

Now if AE = 1

Then EF = the square root of 2 minus 1 =  0.4142

Since EF = BD

Then BD = 0.4142 times the Width of the Octagon

 

 

Noel Reekie's Simplified Proof that the length of the Side of an Octagon is 0.4142 times the Width if the Octagon.

 

AC = 1

AB = 0.5

BO = 0.5

Therefore AO = the square root of 0.5 as does AE, OC and CD = 0.7071

BE = 0.7071 minus 0.5 = 0.2071

Similarly BD also = 0.2071

DE = BE + BD = 0.2071 + 0.2071 = 0.4142

 

Thanks Noel.  Now I can think about something else!