Find the length of the perpendicular to the base of this equilateral triangle.

Geometry Level 2

\(ABC\) is an equilateral triangle with side length 1. Square \(DEFG\) is placed in triangle \(ABC\) such that points \(E,\) \(F,\) and \(G\) are on segments \(AB,\) \(BC,\) and \(CA,\) respectively.

What is the length of the perpendicular from point \(D\) to segment \(BC?\)

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