The answers to parts (i) and (ii) are just silly. An angle will not be the inverse sine of another angle. Part (ii) is not invariant under scaling.
Only part (iii) deserves consideration. To see if the quadrilateral ABDE is always cyclic, one needs to do a calculation. Fix the position and orientation of the figure by putting O2 at the origin and the common point of contact of the two circles at x = -r on the negative x-axis. Then O, the centre of the larger circle (radius R), is on the positive x-axis at x = R-r. Note that O is not on the smaller circle, but a short distance to the right. This means that R > 2r.
I find that once the radius R of the larger circle has been decided, somewhere in the narrow range 2r < R < (2.1796...)r, all the points A, B, G, H, O, I, F and E (and the angle x) are uniquely determined by the constraint HI = IE. But C and D are not determined. The line EDC is free to rotate about its fixed point E. The line ID has a fixed direction, but D itself is free to move up and down this line. This means that D is free to NOT lie on the unique circle through A, B and E. Whenever D is not on that circle, ABDE is not cyclic.
So there you have it.
