query: Inequality of sides of triangle
positive_passages.docid: 23979
positive_passages.text: Inequality involving sides of a triangle
negative_passages.docid: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
negative_passages.docid: 3030
negative_passages.text: Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$
negative_passages.docid: 17264
negative_passages.text: Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 135120
negative_passages.text: Oppenheim's Inequality for triangles, American Mathematical Monthly problems
negative_passages.docid: 137507
negative_passages.text: triangular series and inequality
negative_passages.docid: 137522
negative_passages.text: Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $
negative_passages.docid: 83788
negative_passages.text: Condition on sides of triangle to proof it is isoceles
negative_passages.docid: 69653
negative_passages.text: Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$
negative_passages.docid: 55975
negative_passages.text: Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.
negative_passages.docid: 155423
negative_passages.text: lower triangle inequality for complex numbers
negative_passages.docid: 134764
negative_passages.text: Triangle Inequality question with fractions
negative_passages.docid: 126032
negative_passages.text: Proving a certain inequality
negative_passages.docid: 56954
negative_passages.text: Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero
negative_passages.docid: 120444
negative_passages.text: Trigonometric Triangle Equality
negative_passages.docid: 93599
negative_passages.text: General Proof for the triangle inequality
negative_passages.docid: 150609
negative_passages.text: Triangle Inequality of the Cartesian product with Max function