Bisectors

Geometry Level 4

Let \(ABCD\) be a trapezoid with \(AB//CD\). The bisectors of \(\angle CDA\) and \(\angle DAB\) meet at \(E\), the bisectors of \(\angle ABC\) and \(\angle BCD\) meet at \(F\), the bisectors of \(\angle BCD\) and \(\angle CDA\) meet at G, and the bisectors of \(\angle DAB\) and \(\angle ABC\) meet at \(H\). Quadrilaterals \(EABF\) and \(EDCF\) have areas \(24\) and \(36\), respectively, and triangle \(ABH\) has area \(25\).

The area of triangle \(CDG\) can be expressed as \(\dfrac{m}{n}\), where \(m,n\) are coprime positive integers.

Find \(m+n\).

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