A Geometric Derivation of Pi, Arcsine, and the Inverse Polynomial Expression for Sine

In this paper, we propose a geometric method to derive pi using recursive bisections of chords on a semicircle. We extend this approach to a generic arc to obtain a continuous formula for arcsin(x) via infinite nested radicals, showing its direct connection to the historical infinite product of Francois Viete (1593). Finally, we demonstrate that this geometric process can be completely inverted algebraically. By using successive squarings, we find exact limit formulas for sin(theta) through two alternative polynomial operators, based on external and internal scaling frameworks. In particular, we show that the internal scaling operator directly tracks the physical geometry of the circle by converging to the sagitta of the arc.