This is probably old news, but triangles with two angles in ratio n seem always to have their shortest side a perfect nth power: Sides -> Angle ratios 4 | 5 | 6 -> {0.673407904146284, 2, 0.7424920273751127}, 40 | 39 | 25 -> {1.882031453653618, 1/2, 1.062681495634607}, 48 | 35 | 27 -> {1.363957995353242, 1/3, 2.199481223190493}, 70 | 51 | 49 -> {1.052656745727889, 1/2, 1.899954575047201}, 77 | 72 | 49 -> {1.710412883254859, 1/2, 1.169308311215516}, 117 | 88 | 81 -> {1.113181718197647, 1/2, 1.796651855941546}, 126 | 115 | 81 -> {1.62220636751334, 1/2, 1.232888761906277}, 132 | 119 | 64 -> {2.216537678860817, 1/3, 1.353462216596219}, 176 | 135 | 121 -> {1.153038717449732, 1/2, 1.734547131620664}, 187 | 168 | 121 -> {1.56846197132432, 1/2, 1.275134518123709}, 204 | 145 | 144 -> {1.008861547409698, 1/2, 1.982432579708385}, 228 | 217 | 144 -> {1.779802212058679, 1/2, 1.123720369853131}, 247 | 192 | 169 -> {1.181274044306326, 1/2, 1.693087230384759}, 260 | 231 | 169 -> {1.532276923765853, 1/2, 1.305247092728272}, 273 | 272 | 169 -> {1.982157307044563, 1/2, 1.009001653346091}, 280 | 279 | 125 -> {2.965423277339246, 1/3, 1.011659961977428}, 330 | 259 | 225 -> {1.202325336238882, 1/2, 1.663443279217926}, 345 | 304 | 225 -> {1.506251839762349, 1/2, 1.327799208076356}, 368 | 273 | 256 -> {1.086393099567274, 1/2, 1.840954255689428}, 400 | 369 | 256 -> {1.660214339091134, 1/2, 1.204663731006491}, 425 | 336 | 289 -> {1.21862553465025, 1/2, 1.641193248567542}, 442 | 387 | 289 -> {1.486633544904333, 1/2, 1.345321452522924}, 459 | 440 | 289 -> {1.809233016517494, 1/2, 1.105440803777561}, 476 | 305 | 256 -> {1.216537678860817, 1/4, 3.288019820105906} In[234]:= Times @@ {0.673407904146284`, 2, 0.7424920273751127`} Out[234]= 1. . . . In[233]:= Times @@ {1.2165376788608173`, 1/4, 3.288019820105906`} Out[233]= 1. These 2, 1/2, 1/3, etc, were Rationalize'd after FullSimplify failed utterly on all of them. --rwg