Sreejato's triangle

Geometry Level 5

In triangle ABCABC, AB=41 AB = \sqrt{41} and AC=164 AC = \sqrt{ 164} . The internal angle bisector of BAC BAC meets BCBC at DD. Given that lengths ADAD and DCDC are positive integers, what is the value of AD+DCAD+DC?

This problem is proposed by Sreejato.

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