Triangle Numbers

Take a look at the multiplication square below.

The first eleven triangle numbers $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$ have been identified.

Can you see a pattern?
Does the pattern continue?
If so, can you explain why triangle numbers take up these positions in a multiplication square?




Thank you to Paul Stephenson from The Magic Mathworks Travelling Circus who provided the idea for this problem.