VISUAL THINKING OF FACTORING FORMULAE OF SIMPLE THIRD-ORDER SYMMETRIC EQUATIONS $a^3+b^3+c^3-3 a b c \text { AND }(a+b+c)(a b+b c+c a)-a b c$

Abstract

Visual thinking of the factorisation formulae of simple third-order symmetric equations $a^{3}+b^{3}+c^{3}-3 abc$ and $(a + b + c) (ab + bc + ca) - abc$ is provided by using tiled rectangles.
These proofs are based on an equal-area transformation.
The geometric meanings of $a^{3}+b^{3}+c^{3}-3abc = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - bc - ca)$ and $(a+b+c)(ab+bc+ca)-abc=(a+b)(b+c)(c+a)$ can be expressed by the same diagram.

Received: August 5, 2025
Accepted: August 23, 2025