Trigonometry or something else?

Geometry Level 4

The A-excircle of triangle \(ABC\) is defined to be the unique circle which is tangent to lines \(AB\), \(BC\) and \(AC\) of triangle \(ABC\) which lies on the opposite side of line \(BC\) as \(A\).

Triangle \(ABC\) has side lengths \(BC = 98\), \(AC = 76\), \(AB = 54\), and its A-excircle is tangent to line \(AC\) at \(D\). Find the length of \(CD\) up to three decimal places.

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