Pick an easy case
Pick an easy case and save a lot of work, and use no Trig. Work backward from guess that Bx/By = 2 (it is good guess by GeoGebra).
I love this sort of problem. You are given that there is a ratio. Clearly it is independent of length AB by scaling symmetry. Pick an especially convenient case. Let AB=10 and point B is origin.
Choose these ...
B=(0,0), A=(0,10), C=(10,0)
Guess that
y=(1,0), x=(0,2) and z=(2,3)
Notice vector xy is -1 across and 2 up.
Notice vector zx is 2 across and 1 up.
Proving that zxy is a right angle.
Prove that
z lies on line AC
done