Prime Triangle

Geometry Level 3

The figure above shows an isosceles triangle ABCABC with BA=ACBA = AC and BAC=120\angle BAC = 120^\circ.

Let DD be a point on ACAC such that AD>DCAD > DC. Given that the lengths of BABA, ADAD, DCDC, and BDBD are all distinct prime numbers, what is the length of BD?BD?

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