MathDB

Let

O

be the center of an equilateral triangle

ABC

of area

1/\pi

. As shown in the diagram below, a circle centered at

O

meets the triangle at points

D

E

F

G

H

, and

I

, which trisect each of the triangle's sides. Compute the total area of all six shaded regions. [asy] unitsize(90); pair A = dir(0); pair B = dir(120); pair C = dir(240); draw(A -- B -- C -- cycle); pair D = (2*A + B)/3; pair E = (A + 2*B)/3; pair F = (2*B + C)/3; pair G = (B + 2*C)/3; pair H = (2*C + A)/3; pair I = (C + 2*A)/3; draw(E -- F); draw(G -- H); draw(I -- D); draw(D -- G); draw(E -- H); draw(F -- I); pair O = (0, 0); real r = 1/sqrt(3); draw(circle(O, r)); fill(O -- D -- E -- cycle, gray); fill(O -- F -- G -- cycle, gray); fill(O -- H -- I -- cycle, gray); fill(arc(O, r, -30, 30) -- cycle, gray); fill(arc(0, r, 90, 150) -- cycle, gray); fill(arc(0, r, 210, 270) -- cycle, gray); label("

A

", A, A); label("

B

", B, B); label("

C

", C, C); label("

D

", D, unit(D)); label("

E

", E, unit(E)); label("

F

", F, unit(F)); label("

G

", G, unit(G)); label("

H

", H, unit(H)); label("

I

", I, unit(I)); label("

O

", O, C); [/asy]

Problem Statement