 Triangular number square-complement identity

For $n\ge 1$,An $n$-by-$n$ array of cells has $n^2$ cells. One side of the diagonal, including the diagonal itself, contains $1+2+\cdots +n=T_n$ cells; the other contains $1+2+\cdots +(n-1)=T_{n-1}$. For $n=5$, the two counts are fifteen and ten, giving twenty-five. The identity is stated in Introduction to Arithmetic by Nicomachus (Hahn teacher edition) (Book II, Chapter 12, §§1–2, printed p. 70).