Integer triangle not) satisfy formula_151. We can generate the full equivalence class of similar triangles that satisfy B=3A by using the formulas where formula_28 and formula_27 are integers such that formula_157. Note that all triangles with B = 3A (whether with integer sides or not) satisfy formula_158. The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the equilateral triangle. This is because integer sides imply three rational cosines by the law of cosines, and by Niven's theorem a rational cosine coincides with a rational angle if and only if the
Ideal triangle Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space. In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles. In the Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary