Three congruent circles are inscribed in an equilateral triangle. Show that the distance from a vertex to the closest circle is the circle's radius.

Question

Why is $r$ the length of the circle’s radius? How would I describe the triangle’s side length?

(This image looks a little bit 3D, but it's supposed to be 2D. The $r$ line is just a guide)

Answer 1

Let $R$ be the radius of a circle. Consider the triangle formed by the top vertex of the triangle, the centre of the upper circle and the intersection of this circle and the triangle. It is a right angled triangle with angles of 30 and 60 degrees.

Then $\sin 30 =\frac{1}{2}=\frac{R}{R+r}$ and so $R=r$.

The triangle side is then $(\sqrt 3+2+\sqrt 3)r=2(1+\sqrt 3)r$.