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, \(\overline{xyz}\)?

Image Credit: Harry Potter & the Prisoner of Azkaban.

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