I have worked through proving all the basic properties of the triangle, but I am confused about the concluding statement the author makes after this:
‘The relation R = 2r, which is a consequence of the coincidence of the circumcenter (intersection of perpendicular bisectors) and the incenter (inter- section of angle bisectors) with the centroid (intersection of medians which trisect one another), is but an outer manifestation of the hidden inner rela- tions of Figure 2.1. It immediately implies that the area of the annular region between the circumcircle and the incircle is three times the area of the latter. In turn, this leads directly to the beautiful relation that the sum of the shaded areas equals the area of the incircle!’
I think I understand the first part of his statement, but it is the bit in bold I am unsure about. How do we reach the conclusion that the shaded areas sum to the area of the incircle?
$$ \eqalign{ & A_{\,R} = 4A_{\,r} \cr & {1 \over 3}\left( {A_{\,R} - A} \right) + {1 \over 3}\left( {A - A_{\,r} } \right) = {1 \over 3}\left( {A_{\,R} - A_{\,r} } \right) = \cr & = {1 \over 3}\left( {4A_{\,r} - A_{\,r} } \right) = A_{\,r} \cr} $$