Line in Right Triangle

Consider the circumcircle of triangle $BDE$ as $O$ . As $\angle AEB=\angle ABD$ , we can conclude line $AB$ is the tangent of circle $O$ , which tangential point is $B$ . As line $BC$ is perpendicular to line $AB$ at the tangential point $B$ , we can conclude the circumcenter of triangle $BDE$ lies on line $BC$ .

As $GE=GD$ , point $G$ lies on the perpendicular bisector of line $DE$ . As the circumcenter of triangle also must lies on perpendicular bisector of its side, and also $G$ lies on line $BC$ , we can conclude that $G$ is the circumcenter of triangle $BDE$ . Hence, $BG=GE=\large 30$ .