This Demonstration shows Lill's graphic method for solving a quadratic equation. The value is a root of the equation if and only if . In this case, the point is on a circle with diameter .

Special case: the golden ratio is the positive root of the equation .

Here is a simple construction of the golden ratio. Draw a zigzag line through the points , , , . Let be a circle with radius and center at the midpoint of the segment . Let be the intersection of and an extension of .

The golden ratio is .

Reference

[1] D. Kurepa, Higher Algebra, Book 2 (in Croatian), Zagreb: Skolska knjiga, 1965 pp. 1072–1074.