List questions from geometric-inequalities

Published on 18 Nov 2012 - 23:10

157

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Geometric inequality regarding a tetrahedron

Published on 18 Feb 2013 - 6:33

113

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Using vectors to solve geometric inequality questions

Published on 13 Aug 2020 - 9:55

244

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Is $1-\cos nt \leq n(1-\cos^nt)$?

Published on 13 Aug 2020 - 17:47

158

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Prove $ac\cos B+ab\cos C-bc\cos A-a^2 \le \frac{c^2}{8\cos^2(90^\circ-C)}$ for $\triangle ABC$

Published on 18 Aug 2020 - 17:51

261

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$(a+1)(b+1)(c+1)\leq4$ for triangle sides $a,b,c$ with $ab+bc+ac=1$

Published on 20 Aug 2020 - 10:46

172

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Does this strong convexity estimate hold?

Published on 25 Mar 2026 - 22:09

74

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Given four edge-lengths of a quadrangle $a,b,c,d$ so that $a\leq b\leq c\leq d$. Prove that $a^{2}+b^{2}+c^{2}+d^{2}<2\left(ab+ac+ad+bc+bd+cd\right).$

Published on 25 Aug 2020 - 11:51

96

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Inequality dealing with area and sides of a triangle

Published on 26 Aug 2020 - 13:00

405

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Prove that $a^3 + b^3 + 3abc > c^3$

Published on 13 Sep 2020 - 18:08

140

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To Prove $\frac{1}{b}+\frac{1}{c}+\frac{1}{a} > \sqrt{a}+\sqrt{b}+\sqrt{c}$

Published on 26 Mar 2026 - 21:27

190

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Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .

Published on 25 Mar 2026 - 19:11

64

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Proof that $2(a^4+b^4+c^4)<(a^2+b^2+c^2)^2$ if and only if line segments of length $a$,$b$ and $c$ form a triangle

Published on 23 Feb 2026 - 17:01

69

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In which condition，the triangle have the maximum triangle area？

Published on 28 Oct 2020 - 2:01

118

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Prove this geometric inequality

Published on 29 Oct 2020 - 3:52

List Question

Geometrical Inequalities

Geometric inequality regarding a tetrahedron

Using vectors to solve geometric inequality questions

Is $1-\cos nt \leq n(1-\cos^nt)$?

Prove $ac\cos B+ab\cos C-bc\cos A-a^2 \le \frac{c^2}{8\cos^2(90^\circ-C)}$ for $\triangle ABC$

$(a+1)(b+1)(c+1)\leq4$ for triangle sides $a,b,c$ with $ab+bc+ac=1$

Does this strong convexity estimate hold?

Given four edge-lengths of a quadrangle $a,b,c,d$ so that $a\leq b\leq c\leq d$. Prove that $a^{2}+b^{2}+c^{2}+d^{2}<2\left(ab+ac+ad+bc+bd+cd\right).$

Inequality dealing with area and sides of a triangle

Prove that $a^3 + b^3 + 3abc > c^3$

To Prove $\frac{1}{b}+\frac{1}{c}+\frac{1}{a} > \sqrt{a}+\sqrt{b}+\sqrt{c}$

Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .

Proof that $2(a^4+b^4+c^4)<(a^2+b^2+c^2)^2$ if and only if line segments of length $a$,$b$ and $c$ form a triangle

In which condition，the triangle have the maximum triangle area？

Prove this geometric inequality

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