Can you please show with example of an obtuse angle with unequal sides how external bisectors of an angle can meet the opposite side? unless i curve the plane of paper, this thing is beyond my ability to perceive or understand. I can not prove that they will ever intersect the opposite sides according to Euclidean Geometry.

You can try to use the Exterior Bisector theorem. Or try to find the sum of the angles formed by joining two pairs of intersection points is \(180^{\circ}\)