Constructions Around a Circle

Geometry Level 5

Outside of circle \( \Gamma\), point \(A\) is chosen. Tangents \(AB\) and \(AC\) to \(\Gamma\) are drawn, with \(B, C\) on the circumference of \(\Gamma\). On the extension of \(AB\), \(D\) is a point such that \(B\) is between \(A\) and \(D\), and \( \angle ADC = 25 ^ \circ\). The circumcircle of \(ADC\) intersects \(\Gamma\) again at \(E\) (different from point \(C\) ). \(F\) is the foot of the perpendicular from \(B\) to \(DC\). What is the measure (in degrees) of \(\angle DEF\)?

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