Proof for (1 + 2 + ... + n) = ½n(n+1) shown below.

1) Sums of Alternating Odd Numbers (PDF) by Arthur Benjamin proves .

2) Putnam 2004 Problem (PDF): "Basketball star Shanille O-Keal's team statistician keeps track of the number S(N) of successful free throws she has made in her first N attempts of the season. Early in the season, S(N) was less than 80% of N, but by the end of the season, S(N) was more than 80%. Was there necessarily a moment in between when S(N) was exactly 80% of N?"

3) Roger Nelsen's homepage contains several of his publications on the subject. For example, Area of an Arbelos (PDF) — Salinon (PDF) — Euler's Triangle Inequality (PDF). A wonderful article by Roger Nelsen: Paintings, Plane Tilings and Proofs (PDF) that points out mathematical theorems implicit in tilings!

4) AM-GM inequality by Jim Wilson, University of Georgia.

5) Three proofs without words: (A) , (B) Law of Cosines: , and (C) .

6) An article at cut-the-knot.org traces the history of Proofs Without Words.

7) Infinite Geometric Sums: (A) , and (B) .

8) Proof that 1 + 2 + 3 + ... + n = n choose 2 at Wolfram.

9) Euler's arctangent property by Rex H Wu proves .

10) Geometric series: (A) , and (B) .

11) Proof for Integral: .

12) Lecture Notes.

13) Viviani's Theorem.

Proofs Without Words have appeared in various journals and magazines for decades now. Three books contain collections of such proofs:

1) Q.E.D.: Beauty in Mathematical Proof (64 pages, 2004) by Burkard Polster.

2) Proofs without Words: Exercises in Visual Thinking (160 pages, 1997) by Roger Nelsen.

3) Proofs without Words II: More Exercises in Visual Thinking (142 pages, 2001) by Roger Nelsen.

1) Find a "Proof Without Words" for the following problem: "Given two points A and B on the same side of a line, find point P on the line such that AP + BP is minimized." Solution.

2) Use the above insight to prove that the tangent to an ellipse always reflects a ray of light from one focus to the other. Proofs without words at MathOverflow.Net

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