# Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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### Inequality in an acute-angled triangle

Let $ABC$ be an acute-angled triangle and $M$ a point inside it. We denote by $C_1$, $C_2$, $C_3$ the centers of the Euler circles corresponding to the triangles $BMC$, $CMA$, and $AMB$, respectively. ...
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### Longest diagonal in a convex quadrilateral [closed]

Is is true that, in any convex quadrilateral $\mathcal{Q}$, the longest distance between any two points of $\mathcal{Q}$ is attained when we join two non-adjacent vertices of $\mathcal{Q}$? I suppose ...
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### $ab+bc+ca \le 4{\sqrt 3}\Delta$ for a triangle with sides $a$, $b$ and $c$ with area $\Delta$ [duplicate]

If $a$, $b$ and $c$ are the sides of a triangle with area $\Delta$, prove that $ab + bc + ca \le 4\sqrt3\Delta$ and prove that the equality holds iff the triangle is equilateral. I tried to approch ...
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### $\cos(a)+\cos(b)-\cos(a+b)\geq 1$

I am trying to prove that $$\cos(a)+\cos(b)-\cos(a+b)\geq 1$$ For $a,b \geq 0$ and $0\leq a+b\leq 180^°$ I have checked in Wolfram Alpha that the inequality is true, but I am not able to prove it. The ...
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### Geometric-arithmetic mean inequality applied to eigenvalues

I've applied the arithmetic-geometric mean inequality to the eigenvalues of a positive definite matrix $X$, so $det(X)^{1/n}≤tr(x)/n$. Now I would like to show when equality holds. I already found out ...
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### Distance from vector vs distance from projection

Let $x,y,z,w \in \mathbb{R}^n$. Assume $x^\top w \geq 0$, $y^\top w \geq 0$, $z^\top w = 0$ and $\|x\| = \|y\| = \|z\| = 1$, for a general norm $\|\cdot \|$. Moreover, define $p(x,w)$ to be the ...
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### Euclidean angle vs euclidean distance between two vectors

Let $x,y,z \in \mathbb{R}^n$ and define the euclidean angle between two vectors as $$a(x,y) := \arccos\left(\frac{x^\top y}{\|x\|_2\|y\|_2}\right).$$ Assuming $\|x\| = \|y\| = \|z\| = 1$ for a ...
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### Another proof of Euler's inequality via the half-angle formulas

The Euler's inequality is an immediate consequence of Euler's identity in a triangle, $$OI^2=R^2−2Rr.$$ An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
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### Showing $\sum_{cyc}\tan\frac\alpha2\tan\frac\beta2\geq4$ for a cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with sides $a$, $b$, $c$ and $d$. Denote $s$ the semiperimeter and let $\angle{DAB}=\alpha$, $\angle{ABC}=\beta$, $\angle{BCD}=\gamma$ and $\angle{CDA}=\delta$. ...
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### Isoperimetric Inequalities for Infinite Regular Graphs

Say we have the two-dimensional regular graph $G=(\mathbb{Z}^2,S)$ with generator $S$ such that for every vertex we have $\operatorname{deg}(v)=\lvert S\rvert=\text{const.}$ for all $v\in G$. If we ...
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### Largest possible side of a triangle when $\cos(3P)+ \cos(3Q)+ \cos(3R) = 1$.

In triangle $PQR$, $\cos(3P)+ \cos(3Q)+ \cos(3R) = 1$. Two sides of the triangle have lengths of $15 cm$ and $18 cm$. If the length of the third side of the triangle PQR is $\sqrt{m}$ cm, then the ...
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### Let $h_{1},h_{2},h_{3}$ be the altitudes and $m_{1},m_{2},m_{3}$ be the medians of the triangle ABC.

Show that:$$\frac{h_1}{m_1}+\frac{h_2}{m_2}+\frac{h_3}{m_3}\leq3$$ So, I was wondering if we could prevent all the hefty geometry and solve this using Chebyshev's or the Rearrangement inequality. ...