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area of region enclosed by 3 circles, pairwise tangent, 90-60-30 triangle

Source: Dutch NMO 2010 p1

September 6, 2019
geometrycircles

Problem Statement

Consider a triangle ABCABC such that ∠A=90o,∠C=60o\angle A = 90^o, \angle C =60^o and ∣AC∣=6|AC|= 6. Three circles with centers A,BA, B and CC are pairwise tangent in points on the three sides of the triangle. Determine the area of the region enclosed by the three circles (the grey area in the figure).
[asy] unitsize(0.2 cm);
pair A, B, C; real[] r;
A = (6,0); B = (6,6*sqrt(3)); C = (0,0); r[1] = 3*sqrt(3) - 3; r[2] = 3*sqrt(3) + 3; r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7)); draw(A--B--C--cycle); draw(Circle(A,r[1])); draw(Circle(B,r[2])); draw(Circle(C,r[3]));
dot("AA", A, SE); dot("BB", B, NE); dot("CC", C, SW); [/asy]
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