Second Column and Diagonal of Pascal's Triangle consist of Triangular Numbers

Theorem
The third column and third diagonal of Pascal's triangle consists of the set of triangular numbers.

Proof
Recall Pascal's triangle:

By definition, the entry in row $n$ and column $m$ contains the binomial coefficient $\dbinom n m$.

Thus the $2$nd column contains all the elements of the form $\dbinom n 2$.

The $m$th diagonal consists of the elements in column $n - m$.

Thus the $m$th diagonal contains the binomial coefficients $\dbinom n {n - m}$.

By Symmetry Rule for Binomial Coefficients:
 * $\dbinom n {n - m} = \dbinom n m$

Thus the $2$nd diagonal also contains the binomial coefficients $\dbinom n 2$.

By Binomial Coefficient with Two: Corollary, the triangular numbers are precisely those numbers of the form $\dbinom n 2$.

Hence the result.