Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition

FIGEOMETRY

This chapter explores additional delightful properties of the golden ratio in Euclidean, solid, and analytic geometry. Always look for more occurrences of the sacred ratio. We begin our pursuit with an interesting problem studied by Hunter in 1963 [335].

18.1 THE GOLDEN RATIO AND PLANE GEOMETRY

Example 18.1

1. Locate the points P and Q on two adjacent sides of a rectangle such that the areas of triangles , , and are equal; see Figure 18.1.

   Solution

   Let , and . Since the areas of , , and are equal, we have ; that is, . Equation yields ; that is, . From the equation , we have ...