Triangle \(ABC\) is shown with an inscribed circle of radius 4 units centered at point \(D\). The inscribed circle is tangent to side \(AB\) at the point \(G\). The length of \(AG\) is 6 units and the length of \(BG\) is 8 units. What is the measure of angle \(A\)?

Point \(D\) lies on the angle bisector of angle \(ACB\). Point \(E\) lies on the perpendicular bisector of side \(AB\).

1. What can we say about the distance between point \(D\) and the sides and vertices of triangle \(ABC\)?
2. What can we say about the distance between point \(E\) and the sides and vertices of triangle \(ABC\)?