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In \(\triangle ABC, AB = AC = 10\) and \(BC = 12\). Point \(D\) lies strictly between \(A\) and \(B\) on \(\overline{AB}\) and point \(E\) lies strictly between \(A\) and \(C\) on \(\overline{AC}\) so that \(AD = DE = EC\). Then \(AD\) can be expressed in the form \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p + q\).
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