A geometry problem by Kunal Maan

Geometry Level 5

Circles \(P\) and \(Q\) have radii 1 and 4 respectively and are externally tangent at point \(A\). Point \(B\) is on \(P\) and point \(C\) is on \(Q\) such that \(\overline{BC}\) is a common external tangent of the two circles. A line \(L\) through \(A\) intersects \(P\) again at \(D\) and intersects \(Q\) again at \(E\). Points \(B\) and \(C\) lie on the same side of the line \(L\) and the areas of \(\triangle DBA\) and \(\triangle ACE\) are equal. This common area is \(\frac{m}{n}\), where \(m,n\) are co-prime positive integers. Find the value of \(m+n\).

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