Circles in a circle

Geometry Level 4

Given that there are three identical circles with radius $r$ , each touches each other on their respective circumferences and they are all inscribed in a circle with radius $R$ . If the ratio of $r$ to $R$ can be expressed as

$\large \dfrac1{\frac a{\sqrt b} + c}$

for positive integers $a,b$ and $c$ with $b$ square-free, find the value of $a+b+c$ .

The answer is 6.

1 solution

Ahmad Saad
Nov 16, 2015

Four circles to a kissing come.

The smaller are the benter.

The bend is just the inverse of

The distance from the center.

Though their intrigue left Euclid dumb

There's now no need for rule of thumb.

Since zero bend's a dead straight line

And concave bends have minus sign,

The sum of the squares of all four bends

Is half the square of their sum.

3/r^2 + 1/r^2 = 0.5*(3/r - 1/R)^2

3/r^2 + 1/R^2 = 9/(2 r^2) + 1/(2R^2) - 3/(r R)

Solving we get

r/R = - 3 + 2*sqrt(3) => r/R = 1/(2/sqrt(3) + 1)

Vijay Simha - 3 years ago

Circles in a circle

The answer is 6.

1 solution

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