Triple of Triangular Numbers whose Pairwise Sums are Triangular

Theorem
The following triplet of triangular numbers has the property that the sum of each pair of them, and their total, are all triangular numbers:
 * $66, 105, 105$

Proof
Throughout we use Closed Form for Triangular Numbers, which gives that the $n$th triangular number can be expressed as:
 * $T_n = \dfrac {11 \times 12} 2$

We have:

Then: