query: Trigonometric Property when $a+b+c=abc$
positive_passages.docid: 32197
positive_passages.text: Prove that $\tan A + \tan B + \tan C = \tan A\tan B\tan C,$ $A+B+C = 180^\circ$
negative_passages.docid: 73029
negative_passages.text: If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $
negative_passages.docid: 100907
negative_passages.text: $\sin\alpha + \sin\beta + \sin\gamma = 4\cos{\frac{\alpha}{2}}\cos{\frac{\beta}{2}}\cos{\frac{\gamma}{2}}$ when $\alpha + \beta + \gamma = \pi$
negative_passages.docid: 45775
negative_passages.text: In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$
negative_passages.docid: 115929
negative_passages.text: In a triangle $\angle A = 2\angle B$ iff $a^2 = b(b+c)$
negative_passages.docid: 131168
negative_passages.text: Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 94091
negative_passages.text: In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$
negative_passages.docid: 31575
negative_passages.text: Solving trigonometric equations of the form $a\sin x + b\cos x = c$
negative_passages.docid: 29424
negative_passages.text: Maximum value of $\sin A+\sin B+\sin C$?
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: 6096
negative_passages.text: If $A+B=AB$ then $AB=BA$
negative_passages.docid: 115504
negative_passages.text: Find the value of a+b+c
negative_passages.docid: 79
negative_passages.text: Find $a$, $b$ and $c$ that satisfy the following equations
negative_passages.docid: 50296
negative_passages.text: Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
negative_passages.docid: 118716
negative_passages.text: If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$
negative_passages.docid: 23080
negative_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$