# The Lightning Scar

Geometry Level 3

Long after Voldemort's defeat, Harry Potter, now a specialized and well-known Auror, just discovered that his lightning scar (marked red) was, in fact, a part of another symbol for the fourth Deadly Hallow: the Congruence Spear, which could usurp the magical power from whomever it killed.

The segments $$AC$$, $$CB$$, & $$CE$$ are measured $$4$$, $$2$$, & $$6$$ respectively as shown above, and the new mark(blue) is constructed such that the $$\triangle ABC$$ is similar to the $$\triangle AED$$ with $$AB + BC = BD + DE$$.

If $$AB = x$$; $$BD = y$$; and $$DE = z$$ for some numbers $$x, y, z$$, what is the value of the 3-digit integer, $$\overline{xyz}$$?

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