I love the tactics meant for alluring: "very hard question", "my teacher couldn't answer", "impossible to solve", "many smart people".
Anyway, the vector BG is certainly parallel to the vector AH, which is why BG is some constant multiple of AH = b.
Draw the straight line segment from B to G, and draw the vertical line segment from A to the line from B to G, and call the point of intersection for I. It's not hard to convince yourself that AHGI is a parallellogram, so the lenght of IG is the same as the lenght of b. To find the rest of the lenght of BG, you need to find the distance from B to I, but do notice that triangle ABI has one right angle and two angles of both 45 degrees. Therefore AI has the same lenght as a (which is the same as the lenght of b), and by the Pythagorean theorem, you get that the lenght of BI is sqrt(2) times the lenght of b, in total, the vector BG is (1+sqrt(2))b.