Thread: 3 circles incscribed in one

1. 3 circles incscribed in one

Find T in terms of R

I have found that the three circles with radius T form an equilateral triangle

The answer is supposed to be T=0.464R

2. If we connect the vertices of that equilateral triangle into the center we have the radius point of the larger circle. Then, we create 3 isosceles triangles.

Each of these triangles we can use the law of cosines.

$(2T)^{2}=a^{2}+b^{2}-2abcos(\frac{2\pi}{3})$

But a=b, so:

$4T^{2}=2a^{2}(1-cos(\frac{2\pi}{3}))$

$a=\frac{\sqrt{2}T}{\sqrt{1-cos(\frac{2\pi}{3})}}$

Then radius of the large circle is R, so we have:

$R=\frac{\sqrt{2}T}{\sqrt{1-cos(\frac{2\pi}{3})}}+T$

Solve for T and we have:

$T=(2\sqrt{3}-3)R\approx .464R$

3. Originally Posted by realintegerz

Find T in terms of R

I have found that the three circles with radius T form an equilateral triangle

The answer is supposed to be T=0.464R
As the triangle (formed by the centres of the three inscribed circles) in the attachment is equilateral we have:

$Oc=\frac{2T}{\sqrt{3}}$

and so:

$
R=Oa=T+Oc=T\left(1+\frac{2}{\sqrt{3}}\right)
$