What Is This Line?

Geometry Level 4

In ABC,\triangle ABC, AB=6,BC=7,CA=5.AB= 6, BC= 7, CA= 5. There exists a unique line \ell passing through AA such that reflections of B,CB,C about \ell lie on lines CA,ABCA, AB respectively. Suppose \ell intersects BCBC at point D.D. If BDDC=ab\dfrac{BD}{DC}= \dfrac{a}{b} for some coprime positive integers a,b,a, b, find a+b.a+b.

Details and assumptions
- The reflections of B,CB,C in \ell lie on the lines CA,ABCA, AB respectively. They might not lie on the segments CA,AB.CA, AB.
- In the diagram above, B,CB', C' are the reflections of B,CB,C respectively.
- The diagram shown is not accurate.

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