Intensely Hard Circle Question

Question

Compute the ratio of the radius of the circumscribed circle to the radius of the inscribed circle for the 30-30-120 triangle. Find the ratio.

Answer 1

 This is a challenging question with an often elusive answer!

Answer 2

It is difficult to show this without a diagram.

Consider the triangle to be two 30-60-90 triangles side attached to each other with a common altitude.

Let each of these triangles have sides, 1, √3, 2.

The large triangle therefore has sides 2, 2, 2√3

Locating the center of the circumscribed circle, and drawing line segments to this center from the ends of a side of length 2, you can see that the radius of this circle is R = 2.

 

To find the radius of the inscribed circle, you need to locate the position of the incenter.

Using the formula for incenter position,

r =(2√3 x 1)/ (4 +2√3)

∴ R/r = (4√3 +6) /3 ≈ 4.31