The max luns with that requirements will have one arc being tangent to a side by $R$. The other one will be tangent to the other side by $P$. It depends on the length of the lines wich one will be the outer one, and wich one the inner one.
How to find it?
Find the middle point of $RP$ (call it $X$) and the perpendicular to $RP$ by $X$ (call it center-line).
The inner part of the luns will emerge from a circle having its center on this line (center-line). Find its center intersecting the center-line with the perpendicular to $QP$ by $P$ (longest segment).
The outer part will emerge from a circle having its center on center-line, but intersecting it with the perpendicular to $QR$ by $R$ (shortest segment).
Proof: by finding center-line you are finding all points equidistant to $R$ and $P$. So any circle starting and that line and passing by $R$ will also pass by $P$ (and viceversa). By finding the perpendicular to $QR$ passing by $R$ you find a line containg the center of all circles being tangent to $QR$ by $R$ (and the same goes for $QP$ by $P$).