Triangular Number Pairs with Triangular Sum and Difference

Theorem
The following pairs of triangular numbers have sum and difference which are both triangular:


 * $\left({T_5, T_6}\right)$


 * $\left({T_{39}, T_{44} }\right)$


 * $\left({T_{1869}, T_{2090} }\right)$

Proof
That these are the smallest such pairs can be ascertained by trial by exhaustion.