Twin Incircles

Geometry Level 4

Let \(\triangle ABC\) be a triangle with sides \(\overline{AB} = 3\), \(\overline{BC} = 4\) and \(\overline{AC} = 5\). Extend the line \(\overline{BC}\) up to a point \(D\) so that \(D\) is closer to \(C\) than to \(B\).

This new point \(D\) is a point such that the radius of the incircle of the triangle \(\triangle ACD\) is equal to the radius of the incircle of the triangle \(\triangle ABC\). If the ratio \(\frac{\overline{AD}}{\overline{CD}}\) can be expressed as \(\frac {x} {y}\), \(x\) and \(y\) being coprime, then find the value of \(x - y\)

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