Triplication

Geometry Level 4

In \(\triangle ABC\), \(AC = 4\), \(BC = 5\), and \(AB = 6\), point \(E\) is on \(\overline{AB}\) so that \(AE\) = \(\dfrac{1}{2}EB\).

If the sums of \(CE\) and the measure of the segment from \(E\) to the midpoint of \(\overline{CB}\) can be written as: \[\frac{1}{2}\sqrt{P+Q\sqrt{R}},\] where \(P,Q,R\) are all integers. Find \(P\).

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