Questions tagged [geometric-inequalities]
This is a tag for geometric problems involving inequalities.
446
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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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In the plane there are given vectors $a, b, c, d, $, the sum of which is equal to $0$. Prove the inequality!
In the plane there are given vectors $a, b, c, d, $, the sum of which is equal to $0$. Prove the inequality $$\mid a \mid+ \mid b \mid + \mid c \mid + \mid d \mid \geq \mid a+d \mid + \mid b+d \mid + ...
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Geometric inequalities using sides, radius and semi-perimeter
Prove that:
$$\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}S}+1 \ge 2\cdot\frac{4R+r}{\sqrt{3}p}$$
where $a, b, c$ are the sides of the triangle, $S$ is the area, $R$ is the circumradius, $r$ is the inradius and ...
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Sum of sides of a quadrilateral and sum of its diagonals
A proposition says that the sum of the sides of a quadrilateral is greater than the sum of its diagonals. The proof for this proposition goes as follows:-
In quadrilateral $\mathrm{ABCD,}$
we have in $...
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Transform an inequality system into the input for the simplex algorithm
I have a problem that the simplex algorithm was not discussed in the course, but a sample solution uses the simplex algorithm in order to obtain a Gomory Mixed Integer Cut. The dual problem to my ...
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Magnitude of sum of vectors $\le2$
Let $V_1V_2\cdots V_{2n}$ be a convex inscribed polygon of the unit circle. Let $\mathbf x=\sum\limits_{k=1}^n\overrightarrow{V_{2k-1}V_{2k}}$, prove that $|\mathbf x|\le2$.
I started by setting ...
user1034536
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Prove or disprove the inequality if $a,b,c>0$, $a \geq b+c$.
Prove or disprove the inequality
$$a^2b+a^2c+b^2a+b^2c+c^2a+c^2b \geq 7abc$$ if $$a,b,c>0, a \geq b+c.$$
I thought to use this evaluation:
$$a^2b+b^2c+c^2a \geq 3abc.$$
So we have:
$$a^2b+a^2c+b^2a+...
- 479
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Prove or disprove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1$.
Prove or disprove that the inequality
$$\dfrac{1}{\sqrt{1+x}}+\dfrac{1}{\sqrt{1+y}}+\dfrac{1}{\sqrt{1+z}} \geq 1$$
is valid if $x,y,z$ are positive numbers and $$xyz=1.$$
My solution is:
Let $$x=\...
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Prove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1.$
Is given that $x,y,z$ are positive numbers and $xyz=1$, prove that
$$\dfrac{\dfrac{1}{x}}{\sqrt{z^2+1}}+\dfrac{\dfrac{1}{y}}{\sqrt{x^2+1}}+\dfrac{\dfrac{1}{z}} {\sqrt{y^2+1}}>\sqrt{2}.$$
What have ...
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Proving a simple vector inequality [duplicate]
Can you prove that, if $\vec{x}, \vec{y}$ are real vectors and $\vec{x}$'s elements are nonnegative, then
$$ \sum_i x_i \sum_j x_j y_j^2 \geq \left( \sum_i x_i y_i \right)^2 $$
I thought it followed ...
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Feasibility - When is a set defined by linear inequalities empty (in $\mathbb{R}^n$, with $n \geq 4$)?
Consider the set $\mathcal{S} \subseteq \mathbb{R}^n$, defined by the following
$m > n \geq 4 $ linear inequalities:
$$\begin{cases}
a_{1,1}x_1 + a_{1,2}x_2 + \ldots + a_{1,n} x_n \leq b_1\\
~\...
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Norm inequality for Banach Lattices
Let $X$ be a Banach lattice, $\varepsilon>0$ and $x,y\in S_X$.
If $\||x|\pm y\|\le1+\varepsilon$, then is it true that
$\||x|+|y|\|\le1+f(\varepsilon)$?
Where $f$ is some real function satisfying $\...
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Showing $(4H-2L)e^{\frac{H+L}{k}}-2(H-L)(e^{H/k}+e^{\frac{H+2L}{k}})-L(e^{2H/k}+e^{2L/k})\geq 0$ for $k>0$, $H>0>L$, and $H+L>0$
Let $k>0$, $H>0>L$, and $H+L>0$. Prove that
$$(4H-2L)e^{\frac{H+L}{k}}-2(H-L)(e^{H/k}+e^{\frac{H+2L}{k}})-L(e^{2H/k}+e^{2L/k})\geq 0.$$
I'm trying to prove the statement in the title ...
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Prove that $x\sin A+y\sin B+z\sin C\leqslant \frac{\sqrt{\left( x^2+k \right) \left( y^2+k \right) \left( z^2+k \right)}}{k}$
In triangle $ABC$, let $x,y,z,k>0$, prove that
$$x\sin A+y\sin B+z\sin C\leqslant \frac{\sqrt{\left( x^2+k \right) \left( y^2+k \right) \left( z^2+k \right)}}{k}$$
where $k$ satisfies $\frac{x^2}{x^...
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$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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Convex polygons - For $n>3$, $\sum_{i=1}^{n} \cos\theta_i < \cos\left(\sum_{i=1}^{n} {\theta_i}\right)$
It seems that for any convex polygon $P$ with $n>3$ sides and $n$ interior angles $\theta_i$,
$$\sum_{i=1}^{n} \sin\theta_i > \sin\left(\sum_{i=1}^{n} {\theta_i}\right)$$ $$\sum_{i=1}^{n} \cos\...
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How to prove inequalities in geometry
Outline:
In a triangle $\triangle ABC$ let $\overline{AB}$ be the longest of the three sides. Let $G$ be the centroid of $\triangle ABC$ and $M$ the midpoint of $\overline{AB}$. Furthermore, let a ...
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$\displaystyle \frac{h_a}{l_a} + \frac{h_b}{l_b} + \frac{h_c}{l_c} \leq \frac{\sqrt{3(p^2 + r^2 - 8Rr)}}{2R} + \sqrt{3\frac{2R - r}{2R}}$
For a triangle with standard conventions, prove the inequality in the title, that is:
$$\frac{h_a}{l_a} + \frac{h_b}{l_b} + \frac{h_c}{l_c} \leq \frac{\sqrt{3(p^2 + r^2 - 8Rr)}}{2R} + \sqrt{3\left(1 - ...
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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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Why is the triangle inequality equivalent to $a^4+b^4+c^4\leq 2(a^2b^2+b^2c^2+c^2a^2)$?
Consider the existential problem of a triangle with side lengths $a,b,c\geq0$. Such a triangle exists if and only if the three triangle inequalities
$$a+b\geq c,\quad b+c\geq a\quad\text{and}\quad c+a\...
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A Pedoe-like Geometric Inequality
Problem
Given two triangles $ABC$ and $A'B'C'$ where $a,b,c$ and $a',b',c'$ are the corresponding sides and $F, F'$ denotes the areas of the two triangles. Prove:
\begin{equation}
a^2a'^2+b^2b'^2+c^2c'...
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Geometric inequality in regular pentagon
Let $ABCDE$ a regular pentagon inscribed in a circle of center $O$. Let $P$ an interior point of the pentagon from which we consider parallel line segments to all the sides of the pentagon. We know ...
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Question about a geometric inequality
Question:
Studying some geometric inequalities about arbitrary points, I thought of the following conjecture:
Define triangle $ABC$ and let $M$ be an arbitrary point inside triangle $ABC$. Let $MD \...
user989547
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Sum of distances from interior point to vertices is greater than double the sum of distances to edges
Let $\triangle ABC$, $M\in Int(\triangle ABC)$. Let $MD\perp BC$, $ME\perp AC$ and $MF\perp AB$, $D\in BC$, $E\in AC$, $F\in AB$. Prove that
$$MA+MB+MC\geq 2\cdot(MD+ME+MF).$$
My only idea is to use ...
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Geometric inequality with variable point in a triangle and circumcircles' radii
Problem statement
Let $\triangle ABC$ a triangle and $M$ a point inside it. Let $\mathcal{C_c}$ be the circumcircle of $\triangle MAB$ and $\mathcal{C_a}$ and $\mathcal{C_b}$ similarly defined. Note $...
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How to prove this inequality involving $\tanh$?
Let $a>0, b >0$ and let $0<\alpha<\theta<\pi$.
Prove that
$$\frac{1}{\tanh \left(\frac{1}{\frac{\sin (\alpha )}{a \sin (\theta )}+\frac{\sin (\theta
-\alpha )}{b \sin (\theta )}}\...
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How to show an inequality in an inner product space?
Let $V$ be a real inner product space with inner product $\langle\cdot\,,\cdot\rangle$. For $u,v,w\in V$, how to show the following inequality
$$\langle u,v\rangle \langle u,w\rangle\:\leq\:
\frac{1}{...
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Determining all $(a,b)$ on the unit circle such that $2x+3y+1\le a(x+2)+b(y+3)$ for all $(x,y)$ in the unit disk
In the middle of another problem, I came up with the following inequality which needed to be solve for $(a, b)$ : $$2x+3y+1\le a(x+2)+b(y+3)$$ for all $(x, y)\in\mathbb{R}^2$ with $x^2+y^2\le1.$
Here ...
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Prove that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$ if $a^2+b^2+c^2+abc=4$.
Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2+abc=4$. How do you prove that $\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}\ge a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$?
My Approach: I ...
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In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$
Problem: Given a,b,c are length of triangle. Prove that: $$2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$$
Happy Vietnamese Women's ...
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Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression
Given the triangle ABC. Let BC = a, AC = b, AB = c. Find the minimum value of the following expression:
a) $$P=\frac{4a}{b+c-a} + \frac{9b}{c+a-b} + \frac{16c}{a+b-c}$$
b) $$P=\frac{a^3}{2a+bc} + \...
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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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For an acute angled triangle $ABC,$ if $p=\frac{\sqrt3+\sin A+\sin B+\sin C}{2\sin A\sin B\sin C}$, find the range of $p$
$$ p=\frac{\sqrt3+\sin A+\sin B+\sin C}{2\sin A\sin B\sin C}$$
$\displaystyle \sin A+\sin B+\sin C=4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}$ and
$\displaystyle \sin A\sin B\sin C=8\cos\frac{A}{2}...
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Bounds of an Expression
if $(a,b,c)>0$
and $abc(a+b+c)=3$
Then what can you say about the bounds of
$(a+b)(b+c)(c+a)$ ?
Hint:
My Approach
1.assumed $a\geq b \geq c$
2.Tried to think geometrically
3.used the Hadwiger-...
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Show that $\frac{P_a}{a^2}+\frac{P_b}{b^2}+\frac{P_c}{c^2}\ge\frac{3}{4R}$. When is the equality reached?
Show that $\dfrac{P_a}{a^2}+\dfrac{P_b}{b^2}+\dfrac{P_c}{c^2}\ge\dfrac{3}{4R}$. When is the equality reached?
We're dealing with an acute triangle $ABC$ where $h_a,h_b$ and $h_c$ are altitudes. $P_a,...
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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. ...
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Show that $ \frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \geq \frac{1}{\sqrt{2}}(x+y+z) $
Show that for positive reals $x,y,z$ the following inequality holds and that the constant cannot be improved
$$
\frac{x^2}{\sqrt{x^2+y^2}} + \frac{y^2}{\sqrt{y^2+z^2}} + \frac{z^2}{\sqrt{z^2+x^2}} \...
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Does this class of triangles satisfy a certain geometric inequality?
If $\Delta{}ABC$ is a triangle, call the segment perpendicular to $AB$ and containing $C$ the altitude segment at $C$. In brief my question is the following: is it true that for all $0<\theta<\...
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4
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Given $a,b,c$ are sides of a triangle, Prove that :- $\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$
Given $a,b,c$ are sides of a triangle, Prove that :-
$$\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$$
What I Tried:- I was able to solve the left hand side inequality. From RMS-AM ...
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Prove that if $a,b,c$ are sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$
Prove that, if $a,b,c$ are the sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$.
What I Tried:- Nothing special really came in ...
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Systems of linear homogenous inequalities: getting started
I have a number of questions of varying difficulties related to satisfying largish (as large as possible) systems of linear inequalities. I gather these aren't easy, so I'd be happy to get numerical ...
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Proving a conditional algebraic inequality
Let $x,y \in (0,1)$, and suppose that
$$
x^2-2x+y^2<0.
$$
How to prove that
$$
-x^3-xy^2+4y^2 \ge 0.
$$
holds?
The motivation comes from a certain geometric problem (a bit long to describe here).
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Detect if two elliptic cones overlap
Suppose I have two elliptic cones, both of whose vertices are at the same point. Do the interiors of these cones intersect?
I'm working in normal 3-dimensional Euclidean space.
An elliptic cone can ...
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