# triangle inequality

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## 7 matching pages

##### 1: 1.9 Calculus of a Complex Variable

###### Triangle Inequality

…##### 2: 34.4 Definition: $\mathit{6}j$ Symbol

##### 3: Bibliography G

##### 4: 34.5 Basic Properties: $\mathit{6}j$ Symbol

##### 5: 1.2 Elementary Algebra

*triangle inequality*, …

##### 6: 28.29 Definitions and Basic Properties

##### 7: 23.22 Methods of Computation

In the general case, given by $cd\ne 0$, we compute the roots $\alpha $, $\beta $, $\gamma $, say, of the cubic equation $4{t}^{3}-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent ${e}_{1}$, ${e}_{2}$, ${e}_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is ${c}^{3}-27{d}^{2}>0$, then ${e}_{1}$, ${e}_{2}$, ${e}_{3}$ can be identified via (23.5.1), and ${k}^{2}$, $k_{}^{\prime}{}_{}{}^{2}$ obtained from (23.6.16).

If $$, or $c$ and $d$ are not both real, then we label $\alpha $, $\beta $, $\gamma $ so that the triangle with vertices $\alpha $, $\beta $, $\gamma $ is positively oriented and $[\alpha ,\gamma ]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha $, $\beta $, $\gamma $ are collinear, then we label them so that $\beta $ is on the line segment $(\alpha ,\gamma )$. In consequence, ${k}^{2}=(\beta -\gamma )/(\alpha -\gamma )$, $k_{}^{\prime}{}_{}{}^{2}=(\alpha -\beta )/(\alpha -\gamma )$ satisfy $\mathrm{\Im}{k}^{2}\ge 0\ge \mathrm{\Im}k_{}^{\prime}{}_{}{}^{2}$ (with strict inequality unless $\alpha $, $\beta $, $\gamma $ are collinear); also $|{k}^{2}|$, $|k_{}^{\prime}{}_{}{}^{2}|\le 1$.

Finally, on taking the principal square roots of ${k}^{2}$ and $k_{}^{\prime}{}_{}{}^{2}$ we obtain values for $k$ and ${k}^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2{\omega}_{1}$, $2{\omega}_{3}$ are given by

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2{\omega}_{1}$, $2{\omega}_{3}$), corresponding to the 2 possible choices of the square root.