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February 23rd, 2011, 04:45 AM   #1
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Could anyone help me with this figure?

I am working with one of Hippocrats famous moons.
There is one part of the geometry I dont get, see figure!

I know that $a_1+a_2= A$ but how de we know this?
(a_1 = Area defined by AC and the circle
a_2 = Area defined by BC and the circle
A = Area defined by the line AB and a circle with radius |AD| and center in D)
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 Hippocrates moon.pdf (111.0 KB, 7 views)

February 23rd, 2011, 05:31 AM   #2
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Re: Could anyone help me with this figure?

[attachment=0:3rdvgl6b]lunes.PNG[/attachment:3rdvgl6b]

The radius DB is $\sqrt2$ times the length of the radius OB. Therefore the circle centred at D has twice the area as the circle centred at O.

But if you tilt the diagram 45°, you can see that circular segment labelled A is similar to the segments a1 and a2, so its proportions must vary similarly. Thus A must also have twice the area as either of a1 or a2.
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 lunes.PNG (19.4 KB, 115 views)

 February 23rd, 2011, 07:23 AM #3 Member   Joined: Nov 2010 Posts: 53 Thanks: 0 Re: Could anyone help me with this figure? Ah, I see. Thank you.

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