Integer triangle formula_112 and formula_113 generate the same triple. Unfortunately the two pairs can both be of gcd=3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by formula_27 going only till formula_115. Note that we still need to divide by 3 if gcd=3. The only solution for formula_116 under the above constraints is formula_117 for formula_118. With this additional formula_119 constraint all triples can be generated uniquely. An Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees. Integer triangles
Isosceles triangle near-cancellation between the semiperimeter and side length in those triangles. If the apex angle formula_7 and leg lengths formula_8 of an isosceles triangle are known, then the area of that triangle is: This is a special case of the general formula for the area of a triangle as half the product of two sides times the sine of the included angle. The perimeter formula_10 of an isosceles triangle with equal sides formula_2 and base formula_3 is just As in any triangle, the area formula_5 and perimeter formula_10 are related by the isoperimetric inequality This is a strict inequality for isosceles