Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle and externally tangent to a smaller circle. What is the radius of the smaller circle? Express your answer as a common fraction in simplest radical form.
I made some assumptions that look right but would need to be proven.
but the jist of my thinking is this.
let x be the radius of the smallest circle
the interval joining the centres of the 2 top circles is 2 units long
the distance from each of those radii to the middle of the tiny circle is 1+x
the angle between those two lines is 120 degrees
so now you have an isosceles triangle with sides 1+x, 1+x and 2, with the angle between the congruent sides of 120 degrees.
Now use trigonometry to solve.