Geometry

[Microbasic]

Let's say you have a square whose sides are 8 units long. You also have a circle that is congruent to the square on all four sides. We have ANOTHER circle congruent to the circle at one point and to the square at two points. What is the radius of the smaller circle?

+------------+
|  /      \  |
| /        \ |
|/          \|
|            |
|\          /|
| \        / |
|()\      /  |
+------------+

We need to find the radius of circle ().

[sql_lall]

Radius of big circle = 4
Distance centre to vertex = 4*sqrt(2)

let radius of smaller circle be R
=> 4*sqrt(2) = 4 + R + R*sqrt(2)
=> 4*(sqrt(2)-1) = R*(sqrt2+1)
=> 4 *(sqrt(2)-1)/(sqrt(2)+1) = R

=> R = 0.68629... (I think, but i only used a calc. with no bodmas, so it is unreliable ) It may be wrong, so you probably should check.

[DavidHooper]

Verified.
The exact figure is 4 (sqrt(2) -1)^2.

[sql_lall]

Cool, i'd never thought before, but:
(sqrt(2) - 1)/(sqrt(2) + 1) = (sqrt(2) - 1)^2

i.e. (sqrt(2) - 1)^-1 = (sqrt(2) + 1)

[DavidHooper]

Hmm, a useful case of rationalising the denominator. Worth remembering.