Plotting an arc with no center point - a practical solution please! I need a mathematical solution to a very practical problem (laying a patio).
The attached will hopefully explain.
The center of the circle for the arc we wish to have is inaccessible (ie in the house behind walls). 
There are 2 fixed points the arc must intersect.
There is an existing arc drawn using these 2 fixed points, but this circle is too small - radius 539cm
The desired arc is more shallow so would have a greater radius.
I think I need 10 or so measurements from the center of this existing circle, to the new arc. 
These points can then be plotted on the ground and the dots joined up (obviously more than 10 will me more accurate, but 10 seems a sensible number) 
I am sure this is possible, but my mathematical knowledge is not good enough, sorry. Answers or a simple formula I can apply would be very much appreciated.

Anyone out there to help?

If you can measure tangent angles in the diagram the following could be of use in a scaled geometrical construction:
$$ R_{old}= \dfrac{r_2^2-r_1^2}{ 2(r_2 \sin \beta - r_1 \sin \alpha )}$$
Along perpendicular bisector of given points $(1,2)$ mark old center of circle $O$ with this radius and a new center $N$ with a new desired (increased/reduced) radius passing through $(1,2)$ on bisector. Sketch not to scale.
The above result for Irregular Pie is obtained by eliminating invariant semi-chord length $L$ from Circle property that product of two line segments is constant. 
$$ r\cdot (2 R \sin \theta -r)= L^2 $$

EDIT1
Yet another practical way to find $R$ is the relation $ R = \dfrac{s}{\alpha_ d +\beta_d} $ for Diagon from the Gauss-Bonnet theorem. Arc length $s=1P2 $ and corner angles are to be measured.
