Relation between side and radius of circle without any trigonometric functions

Question

We have an equilateral triangle with 3 circles of equal radii inscribed in it. How do we find a relation between the side length "a" of the triangle and radius "r" of the circles? With trigonometric knowledge, mainly 30-60-90 triangles, the question is pretty straightforward, but how does one go about it purely from a geometric viewpoint. I got my answer as a = 2r(1 + $\sqrt{3}$). But have no clue about how to approach the problem without trig.

Answer 1

Using $30°-60°-90°$ triangles:

$$2\sqrt 3r+2r=a$$

$\hspace{3cm}$

"Purely Geometric"

Since the given figure is an equilateral triangle, you will need to use the $60°$ angles somehow. Let's assume you don't know any trigonometry. Let's prove the side length ratios of $30-60-90$ triangles purely geometrically.

Suppose we have a $30-60-90$ triangle with only $HC$ equal to $k$ (in the below figure). When you stick a copy of this triangle next to the original you form an equilateral triangle (all angles are $60°$). Therefore, we know $AC=2k$ and we can apply Pythagorean Theorem in $AHC$ to obtain $AH= r \sqrt3$.

$\hspace{3cm}$