Earlier ellipsis
scaled up by 85 should have read "rescaled to shortest side 12x85" ; the following claim This property is not shared by (for instance) the {3, 4, 5}-sided triangle. is likewise rubbish --- by similar triangles, taking vertices [0,0], [0,5], [12,9]/5 sets the triangle with long side axially aligned.
Finally, the set of transformations considered "trivial" in this context plainly should include rational translations, reflections in axes and their bisectors, and rotations through right-angles. However, any rotation with matrix [[p,q],[-q,p]]/r with p^2 + q^2 = r^2 (Pythagorean triplet) also always conserves rationality. At which point one begins to suspect any pair of congruent rational triangles is related via a rational isometry, though it's not immediately obvious to me how to prove this. Apologies for thoroughly misconscrewed question! WFL On 10/4/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
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