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Dutch Mathematical Olympiad
2010 Dutch Mathematical Olympiad
1
1
Part of
2010 Dutch Mathematical Olympiad
Problems
(1)
area of region enclosed by 3 circles, pairwise tangent, 90-60-30 triangle
Source: Dutch NMO 2010 p1
9/6/2019
Consider a triangle
A
B
C
ABC
A
BC
such that
∠
A
=
9
0
o
,
∠
C
=
6
0
o
\angle A = 90^o, \angle C =60^o
∠
A
=
9
0
o
,
∠
C
=
6
0
o
and
∣
A
C
∣
=
6
|AC|= 6
∣
A
C
∣
=
6
. Three circles with centers
A
,
B
A, B
A
,
B
and
C
C
C
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("
A
A
A
", A, SE); dot("
B
B
B
", B, NE); dot("
C
C
C
", C, SW); [/asy]
geometry
circles