In an equilateral triangle, the internal angle bisectors, 
the altitudes and the medians are all the same. The center
of the inscribed circle is where the angle bisectors cross,
so we draw an angle bisector to the center of the circle,
and a radius from the center of the circle to the lower side
of the triangle.  Since the internal angles of an equilateral
triangle are 60°, the angle bisector of the angle divides it
into two 30° angles. Since each side of the triangle is 2V3, the
raidus divides the bottom side side into two parts each V3 in 
length.  I will label the radius r:



That is a 30°,60°,90° triangle so its hypotenuse is twice the
shorter side,  So the hypotenuse is 2r and we have, by the
Pythagorean theorem:











So the radius is 1.

Edwin