Solving an AAA triangle using circles

Question

I was asked to find the angles of an isosceles triangle by circumscribing a circle around it, then drawing another circle with the tip of the triangle as its center, and a radius of half of the leg.

I can see that the ratio of AD to DE is the same as AB to BC, but how does this help me know the angles of the triangle? The lengths are 10 for the legs and 6 for the base. I know I can use other methods to find the angles, but how do I use this method - meaning, how can I specifically use the circles to find the angles? I know the radius of the upper circle to be $\frac{1}{2} AB$, and that $DE=\frac12 BC$.

Answer 1

Hint. In the upper circle you have the information you need to find the $\sin$ of half the central angle.

Answer 2

Is this what they were asking for?

$\alpha = m \angle BAC = m \operatorname{arc} DE$

$\dfrac{\pi - \alpha}{2} = m \angle ABC = \dfrac 12(m \operatorname{arc} GE)$

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