This paper presents a self-contained proof of Fermat's Last Theorem, showing that for any exponent greater than two, no positive whole numbers satisfy the equation. The proof interprets any hypothetical solution geometrically as the sides of a triangle, applying the Law of Cosines to derive a strict condition involving an angle. This condition is expanded using the Binomial Theorem, revealing a combinatorial inequality that depends on the exponent. For exponents greater than two, this inequality contradicts the requirement that all three numbers be positive integers. A systematic case analysis of the possible ratios between the numbers resolves each possibility using elementary properties of binomial functions and basic inequalities. The entire proof avoids deep theories of elliptic curves and modular forms, using only geometry, algebra, and simple analytic reasoning. The key insight is that for exponents one and two, the geometric relationship holds perfectly, but for any larger exponent, it forces an impossible condition on the triangle's angle, breaking the requirement that all three sides be positive whole numbers.