@Token Bird
If we assume that it has not become stationary, but has continued to fly but has been present on the same arc at both 07:11 and 08:11, can we make the maths work if we know the changing direction of the southern arc if we follow it south and we assume that the aircraft is flying on a constant track? IE. Where would they have converged, as it were? Or would that not have occurred at all?
You'd have to make an assumption about its speed - strictly its ground speed, since the arcs are defined relative to a point in space fixed relative to the Earth's surface (the IOR satellite's geostationary position) . If you did that, then there'd only be one heading that could link any two points on the arc. But since you don't know where the first point is, it doesn't help find the second.
Really you need to start from a known position - last radar plot, for example - and then work out a tree of possible locations each hour from knowing which "arcs" it was on each hour, what airspeed it might have maintained and factoring in winds aloft etc. Horribly complicated and error-prone, but I imagine that there have been clever, well-informed, people doing that for days. It may be how the NTSB tracks on the Aussie search area maps were derived, I suppose.