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MATH / Three unit circles are drawn so they are mutually tangent, as shown below. A…

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problem

Three unit circles are drawn so they are mutually tangent, as shown below. A blue circle that is externally tangent to all three unit circles is drawn. Finally, three red circles are drawn, so that each red circle is externally tangent to two unit circles and externally tangent to the blue circle. Then the radius of each red circle can be expressed in the form a−bcd,\frac{a - b \sqrt{c}}{d},where a,a, b,b, c,c, and dd are positive integers, when simplified. Find a+b+c+da + b + c + d. [asy] unitsize(2 cm); pair A, B, C, D, E, F, O; real s = 2/sqrt(3) - 1, r = (9 - 4*sqrt(3))/33; A = 2/sqrt(3)*dir(150); B = 2/sqrt(3)*dir(30); C = 2/sqrt(3)*dir(270); O = (0,0); D = (r + s)*dir(330); E = (r + s)*dir(210); F = (r + s)*dir(90); filldraw(Circle(O,s),blue); filldraw(Circle(D,r),red); filldraw(Circle(E,r),red); filldraw(Circle(F,r),red); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(C,1)); [/asy]
Plain-text mathematical notation (without MathML)
Three unit circles are drawn so they are mutually tangent, as shown below.  A blue circle that is externally tangent to all three unit circles is drawn.  Finally, three red circles are drawn, so that each red circle is externally tangent to two unit circles and externally tangent to the blue circle.  Then the radius of each red circle can be expressed in the form
(a−b√(c))/(d),where a, b, c, and d are positive integers, when simplified.  Find a+b+c+d.

[asy]
unitsize(2 cm);

pair A, B, C, D, E, F, O;
real s = 2/sqrt(3) - 1, r = (9 - 4*sqrt(3))/33;

A = 2/sqrt(3)*dir(150);
B = 2/sqrt(3)*dir(30);
C = 2/sqrt(3)*dir(270);
O = (0,0);
D = (r + s)*dir(330);
E = (r + s)*dir(210);
F = (r + s)*dir(90);

filldraw(Circle(O,s),blue);
filldraw(Circle(D,r),red);
filldraw(Circle(E,r),red);
filldraw(Circle(F,r),red);
draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(C,1));
[/asy]
Original LaTeX notation
Three unit circles are drawn so they are mutually tangent, as shown below.  A blue circle that is externally tangent to all three unit circles is drawn.  Finally, three red circles are drawn, so that each red circle is externally tangent to two unit circles and externally tangent to the blue circle.  Then the radius of each red circle can be expressed in the form
\[\frac{a - b \sqrt{c}}{d},\]where $a,$ $b,$ $c,$ and $d$ are positive integers, when simplified.  Find $a + b + c + d$.

[asy]
unitsize(2 cm);

pair A, B, C, D, E, F, O;
real s = 2/sqrt(3) - 1, r = (9 - 4*sqrt(3))/33;

A = 2/sqrt(3)*dir(150);
B = 2/sqrt(3)*dir(30);
C = 2/sqrt(3)*dir(270);
O = (0,0);
D = (r + s)*dir(330);
E = (r + s)*dir(210);
F = (r + s)*dir(90);

filldraw(Circle(O,s),blue);
filldraw(Circle(D,r),red);
filldraw(Circle(E,r),red);
filldraw(Circle(F,r),red);
draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(C,1));
[/asy]

level

Level 5

type

Precalculus

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Source and history

task_id
53603063-6407-5c28-9565-c4583d2f0bae
task_key
precalculus--test--53603063-6407-5c28-9565-c4583d2f0bae
task_revision_id
1
config / split
precalculus / test

Official source

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