Playing with Pythagorean Triplets!

Geometry Level 5

A Primitive Pythagorean Triangle (PPT) is a right triangle whose sides have lengths which are integers, and have a greatest common divisor of 1. Among all pairs of non-congruent PPTs which have congruent incircles with an integer radius, let the pair for which this radius is minimized be \((a_1, a_2, a_3)\) and \((b_1, b_2, b_3)\), where \(a_1 < a_2 < a_3\) and \(b_1 < b_2 < b_3\). Find the value of \(a_1 + b_1 + a_2 + b_2 + a_3 + b_3\)?

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