A circle has two radii meeting at 60 degrees.

Question

A smaller circle internally tangent to the original circle and is tangent to the two radii. Find the ratio of the length of the radius of the original circle to the length of the radius of the smaller circle.

Answer 1

Let's start with the inner circle and assign it a radius of 1. Its center lies on the angle bisector of the 60^o angle which means the triangle formed by the center of the large triangle, the center of the small triangle, and the point of tangency between the smaller circle and one of the radii of the large circle is a 30:60:90 triangle, Thus the leg opposite the 30^o is 1 (the radius of the inner circle) and the side opposite the 90^o (the distance between the two centers) is 2.

The distance from the center of the large circle to the center of the small circle is 2, and the distance from that center of the smaller circle to the point of tangency between the two circles is 1, thus the radius of the large circle is 2+1= 3.

Original/smaller = 3/1