Both circles have the same radius: r = 1. If area A = area B = area C, what is the distance between the centers of the circles?

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