Proof of the Perpendicular Bisector Theorem for Chords Using Symmetry and Rotational Invariance

This paper presents an original, fully rigorous proof of the classical theorem stating that the line segment from the center of a circle to the midpoint of any chord is perpendicular to that chord. Developed by the author at age 11, this approach utilizes reflective symmetry and rotational invariance to provide a purely visual-geometric alternative to traditional coordinate-based or congruent-triangle methods. The research is motivated by the author's desire to understand this fundamental geometric property through visual reasoning, challenging the premise that advanced algebraic equations are required to demonstrate the theorem. This work serves as an educational resource for young mathematicians, geometry students, and educators interested in intuitive, algebra-free proofs.