Theorem II: For an equilateral triangle, the ratio of the areas of the circumscribed (outer) and inscribed (inner) circles is 4:1. The area of the ring between the circles is 3 times the area of ...

The probability that a random chord in a circle is longer than the side of an equilateral triangle inscribed in the circle is proved to be 1⁄3 (top), 1⁄2 (middle) and 1⁄4 (bottom).