query: Prove that $r^2+ r_1^2+ r_2^2 +r_3^2=16R^2-a^2-b^2-c^2$
positive_passages.docid: 156596
positive_passages.text: Prove $r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$ for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, exradii $r_1$, $r_2$, $r_3$
negative_passages.docid: 148744
negative_passages.text: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 12949
negative_passages.text: Inductive Proof of Euler's Formula $v-e+r=2$
negative_passages.docid: 152039
negative_passages.text: Prove that $a^{2^n}=1 \mod 2^{n+2}$
negative_passages.docid: 18203
negative_passages.text: Diophantine equation $a^2+b^2=c^2+d^2$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 94580
negative_passages.text: Prove that, $\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$
negative_passages.docid: 44705
negative_passages.text: Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
negative_passages.docid: 126018
negative_passages.text: Show that $ e^{A+B}=e^A e^B$
negative_passages.docid: 44119
negative_passages.text: Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
negative_passages.docid: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 49548
negative_passages.text: If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
negative_passages.docid: 101872
negative_passages.text: Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
negative_passages.docid: 6621
negative_passages.text: Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
negative_passages.docid: 111295
negative_passages.text: Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
negative_passages.docid: 95852
negative_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$