# What Is This Line?

Geometry Level 4

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

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

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