Circles in an Equilateral Triangle 
In the diagram below is an equilateral triangle with side length of 1 unit. $C_1$, $C_2$ and $C_3$ are circles inside the triangle tangent to each other and the sides of the triangle. Find the radius of each circle.


Firstly, I'm inclined to think that the circles must all be equal in size, but I'm not sure how to prove that. And I also tried making a smaller triangle inside the outer triangle by connecting the radii of the circles, but I'm not sure how to proceed after that (or if I'm even on the right track for that matter). 

By construction you have $$1=\sqrt 3r+r+r+\sqrt 3r\Rightarrow r=\frac{\sqrt3-1}{4}\approx 0.183012701$$ Later add an explanatory figure.

