In the above image , \(G\) is the centroid of \(\Delta ABC\) and \(AB=5 , AC=6 , BC=7\) .Let \(m\angle BAC = A\) and \(m\angle BGC=\alpha\). If the relationship between the cosines of angles can be expressed as : \[\cos A - \dfrac{49}{15} = \dfrac{\eta_2\sqrt{\eta_3}\cos\alpha}{\eta_1}\] where \(\eta_{i}\) is a positive integer \(\forall i \in \{1,2,3\}\) , \(\eta_3\) is square free and \(gcd(\eta_1,\eta_2)=1\)

Find the value of \(\eta_1 + \eta_2 + \eta_3 +49\).

On the similar lines , we know:

If you see the above image the angle subtended by side \(BC\) on circumcenter \(O\) is twice the measure of \(\angle A\) subtended on circumference of the circumcircle.

If you see the above image , if \(I\) is the incenter , then the relationship between the angles subtended by \(BC\) is \(\angle BIC = 90^\circ + \dfrac{\angle A}{2}\).

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