# Angle Relationship

Geometry Level 5

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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