Visual Proofs

Interactive geometric representations of famous theorems.

Parameters

Proof by rearrangement: Place 4 congruent right triangles inside a square of side (a+b). The inner square (rotated) has side c. Area equation: (a+b)² = 4·(½ab) + c² ⟹ a²+b²=c². Over 370 proofs exist.
Visual proof: Three copies of the staircase (k² blocks at height k) fit together to form a rectangle of dimensions n × (n+1) × (2n+1)/6. Equivalently proven by induction or telescoping.
Visual proof (r=1/2): Divide unit square. First half = 1/2, half of remainder = 1/4, etc. After infinitely many steps, area = 1. S = 1/(1−r): multiply both sides by (1−r): S−rS = 1.
e^{iπ} + 1 = 0: At θ=π, point reaches (−1, 0i). Hence e^{iπ} = −1 ⟹ e^{iπ}+1=0. Connects e, π, i, 1, 0. Taylor series: e^{iθ} = Σ(iθ)ⁿ/n! = (1−θ²/2!+θ⁴/4!−...) + i(θ−θ³/3!+...) = cosθ + i sinθ.
Proof: The diameter of the circle = a+b. The height at the junction = √(ab) (geometric mean, by the altitude theorem). The radius = (a+b)/2 (arithmetic mean). Since radius ≥ height: AM ≥ GM. Equality iff a = b.