Rich has pointed out that any triangle with rational sides and vertices has rational incentre; in fact the same is true for e-centres, circumcentre, orthocentre, symmedian point. These follow trivially from the expressions in homogeneous barycentric coordinates for cases (1)--(5), (8) given earlier under the thread "Update on IMO/RKG-inspired conics thru' 6 points of triangle". But in the meantime I've also noticed that my pet {12, 17, 25}-sided triangle has another curious property: its coordinates may be taken as [0,0], [0,12], [15,20] or [0,0], [0,25], [36,77]/5 or [0,0], [0,17], [180,385]/17 --- that is, scaled up by 85, any chosen edge may lie along the y-axis, and the coordinates remain integer. This property is not shared by (for instance) the {3, 4, 5}-sided triangle. Has this triangle any further essentially distinct rational poses? What triangles possess multiple nontrivial poses rational on the square lattice? What about polygons? Fred Lunnon