## A triple (a, b, c); T_{A} + T_{B} = T_{C}

Let   $(A, \; B, \; C)$   be a set of three positive integers for which   $T_{A} + T_{B} = T_{C}$

Here’s the smallest triangular number of the form   $A + B + C$   where   $(A, \; B, \; C)$   is a triangular triple:

$(A, \; B, \; C) \; = \; (14, \; 18, \; 23)$

$(14\times 15)/2 \; + \; (18\times 19)/2 \; = \; 276 \; = \; T_{23}$

$14 \; + \; 18 \; + \; 23 \; = \; 55 \; = \; T_{10}$

Find the next triples.

Are there infinitely many such triangular numbers   $A + B + C$   ?

Is it possible for the three numbers of a triangular triple to each be triangular?