Bisecting an Angle

The diagram below demonstrates how to bisect an angle.
Given the two lines that form the angle by crossing, first a circle is drawn at their intersection (the large circle). Then the points where that circle intersects the two sides of the angle are marked, and a circle is drawn around one of them. This circle must be large enough to go at least half way through the angle. Then a circle of the same size is drawn around the second point that was marked earlier. When using the Geogebra software, I copied the circle and then released it at that point, this preserved the size. It is important that these two circles be of the same size. Now select one of the two points where these two circles intersect. Mark that point and construct a line through the point at which the two sides of the angle meet and the point where the two circles intersect.