civonamzuk
May 21, 2015

#1**+10 **

I'll give this one a try....these are always interesting......Each circle has an area of pi....so the middle area must be pi/2

See the following pic....

Let circle A be centered at (0, 0)

We need to first solve this.....

Area of sector ABECA - area of triangle ABC = pi/4 so we have

(1/2)[Θ - sin(Θ)] = pi/4.......with a lttle help from WolframAlpha...Θ = about 2.30988 rads = about 132.3463751816785913° ...and (1/2) of this = 66.17318759083929565°

So the point B is given by :

[ cos(66.17318759083929565°), sin (66.17318759083929565°) ] = (0.403973421087, 0.914770722671)

And by symmetry, circle B is centered at twice the distance from A to G = point D =

( 2*.403973421087, 0) = ( 0.807946842174 , 0 )

So the distance between the circles' centers is just 2AG = AD = about .808

Proof.....area between segment BC and arc BEC of circle A =

Which is about pi/4.......and by symmetry....this is the same area between segment BC and arc BFC of circle

B.......so 2(pi/4) = pi/2

CPhill
May 21, 2015

#1**+10 **

Best Answer

I'll give this one a try....these are always interesting......Each circle has an area of pi....so the middle area must be pi/2

See the following pic....

Let circle A be centered at (0, 0)

We need to first solve this.....

Area of sector ABECA - area of triangle ABC = pi/4 so we have

(1/2)[Θ - sin(Θ)] = pi/4.......with a lttle help from WolframAlpha...Θ = about 2.30988 rads = about 132.3463751816785913° ...and (1/2) of this = 66.17318759083929565°

So the point B is given by :

[ cos(66.17318759083929565°), sin (66.17318759083929565°) ] = (0.403973421087, 0.914770722671)

And by symmetry, circle B is centered at twice the distance from A to G = point D =

( 2*.403973421087, 0) = ( 0.807946842174 , 0 )

So the distance between the circles' centers is just 2AG = AD = about .808

Proof.....area between segment BC and arc BEC of circle A =

Which is about pi/4.......and by symmetry....this is the same area between segment BC and arc BFC of circle

B.......so 2(pi/4) = pi/2

CPhill
May 21, 2015