Geometric Position of Points with Fibonacci and Lucas Numbers Coordinates

Points with Fibonacci and Lucas number coordinates have certain known geometric positions. To date, only a few results on this topic are known, mainly concerning positions on circles and hyperbolas. We present positions on many new conics. Our results are derived from various sources: Fibonacci and Lucas number identities, Diophantine equations, and Kimberling’s results on Fibonacci hyperbolas. We establish new identities for generalized Fibonacci and Lucas numbers, which lead to statements about the geometric positioning of points whose coordinates are fractions of these numbers. Thus, placement on conics is shared not only by the Fibonacci and Lucas numbers, but also by the entire large class of numbers satisfying linear two-term recurrences with fixed coefficients.