I am having trouble with this question - I have made some progress though.
Let ABC be an equilateral triangle. Let X and Y be circles with centres A and B, respectively, and radii both equal to AB. Let D be the point here the tangent to X at C meets Y for a second time. Extend segment CA to meet X for a second time at E. Let Z be the circle with centre D and radius DE and extend EC to meet Z at F. Finally, let G be the midpoint of CF.
Prove that line BG is tangent to X.
However, when I keep proving it, I end up with a reverse case - that is, the tangent condition is only satisfied if I assume something is proven. I tried using cosine rule, but I do not think that that is the method. Any help is appreciated. I have attached my diagram below.
