Triangular Number Pairs with Triangular Sum and Difference

Theorem
The sequence of pairs of triangular numbers whose sum and difference are also both triangular begins:
 * $\tuple {15, 21}, \tuple {105, 171}, \tuple {378, 703}, \tuple {780, 990}, \tuple {1485, 4186}, \tuple {2145, 3741}, \tuple {5460, 6786}, \tuple {7875, 8778}$