## Triangular Num3ers: T(a)+T(b),T(a)+T(c),T(b)+T(c),T(a)+T(b)+T(c)

Find positive integers   $a, \; b, \; c$   such that

$T_{a} \; + \; T_{b}$
$T_{b} \; + \; T_{c}$
$T_{c} \; + \; T_{a}$
$T_{a} \; + \; T_{b} \; + \; T_{c}$

are all triangular numbers.

Here are some solutions:

$T_{11} \; + \; T_{14} \; = \; 171 \; = \; T_{18}$
$T_{11} \; + \; T_{14} \; = \; 171 \; = \; T_{18}$
$T_{14} \; + \; T_{14} \; = \; 210 \; = \; T_{20}$
$T_{11} \; + \; T_{14} \; + \; T_{14} \; = \; 276 \; = \; T_{23}$

$T_{230} \; + \; T_{741} \; = \; 301476 \; = \; T_{776}$
$T_{230} \; + \; T_{870} \; = \; 405450 \; = \; T_{900}$
$T_{741} \; + \; T_{870} \; = \; 653796 \; = \; T_{1143}$
$T_{230} \; + \; T_{741} \; + \; T_{870} \; = \; 680361 \; = \; T_{1166}$

$T_{609} \; + \; T_{779} \; = \; 489555 \; = \; T_{989}$
$T_{609} \; + \; T_{923} \; = \; 612171 \; = \; T_{1106}$
$T_{779} \; + \; T_{923} \; = \; 730236 \; = \; T_{1208}$
$T_{609} \; + \; T_{779} \; + \; T_{923} \; = \; 915981 \; = \; T_{1353}$

$T_{714} \; + \; T_{798} \; = \; 574056 \; = \; T_{1071}$
$T_{714} \; + \; T_{989} \; = \; 744810 \; = \; T_{1220}$
$T_{798} \; + \; T_{989} \; = \; 808356 \; = \; T_{1271}$
$T_{714} \; + \; T_{798} \; + \; T_{989} \; = \; 1063611 \; = \; T_{1458}$

$T_{1224} \; + \; T_{1716} \; = \; 2222886 \; = \; T_{2108}$
$T_{1224} \; + \; T_{3219} \; = \; 5932290 \; = \; T_{3444}$
$T_{1716} \; + \; T_{3219} \; = \; 6655776 \; = \; T_{3648}$
$T_{1224} \; + \; T_{1716} \; + \; T_{3219} \; = \; 7405476 \; = \; T_{3848}$

Find other solutions.

Can you find an infinite family of solutions?