Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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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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Possible values of $\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}$ in a triangle [duplicate]

Let $a,$ $b,$ $c$ be the sides of a triangle. Find the set of all possible values of $$\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}.$$ I tried using the triangle inequality to get $\dfrac{a}{b ...
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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 \...
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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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Lower bound on inradius in terms of diameter of isotropic convex body

Consider a convex body whose tensor of inertia is a multiple of the identity matrix. This means that the body is in some appropriate sense balanced or isotropic. Can one lower bound the inradius (...
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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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If the projection of a geometric inequality is true on each line of the plane, is the inequality true?

The inequality involves only sums and products of the segments. The formalized form of the question from the title is as follows: Let $\{a_{ij}\mid 1\le i\le m, 1\le j\le n\}$, $\{b_{ij}\mid 1\le i\...
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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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$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$

Problem: Prove that in any triangle ABC, the following inequality is true: $$(2a^2+bc)(2b^2+ca)(2c^2+ab)\ge(2a^2+2b^2-c^2)(2b^2+2c^2-a^2)(2c^2+2a^2-b^2)$$ Anyone can help me? I tried AM-GM for right ...
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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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4 votes
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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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5 votes
3 answers
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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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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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Relation between measures of a subset and its boundary

Let $\Omega \subset \mathbb R^n$ a bounded and Lebesgue-measurable set, regular enough to say that its boundary $ \partial\Omega$ is a hypersurface in $ \mathbb R^n$ (for n$\ge 3$). Assuming $\pi_i$ ...
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Is the "triangle inequality for reciprocals" a sufficient condition for altitudes of a triangle?

In connection with a question asked in chat, I thought about this: Question: What are sufficient and necessary conditions for three numbers $h_a$, $h_b$, $h_c$ to be heights of a triangle? I learned ...
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generalized f-mean/power mean inequalities

I'd like to prove the following inequality (which seems to be true by numerics) $$ (p-1)\frac{a^2+b^2}{2} \leq \Big(\frac{a^p+b^p}{2}\Big)^{2/p} $$ for all $a,b\in [0,1]$ and $p\in [1,2]$. I'd ...
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Prove that $AG+GB+GH+DH+HE\ge CF.$

In the diagram, $ABCDEF$ is a hexagon with $AB=BC=CD$ and $DE=EF=FA$. Angles $BCD$ and $EFA$ both equal $60°$. $G$ and $H$ are two points taken from inside the hexagon such that angles $AGB$ and $DHE$ ...
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Inequalities between areas of triangles in an Euclidean 4-simplex?

The 10 edge lengths of a 4-simplex with vertices $0,1,2,3,4$ satisfy the inequality that the eigenvalues of the matrix $M$ with elements $$M = [m_{ij}] = l_{0i}^2+l_{0j}^2-l_{ij}^2$$ is positive ...
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4 answers
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Inequality $a^2b(a-b) + b^2c(b-c) + c^2a(c-a) \geq 0$

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that: $$a^2b(a-b) + b^2c(b-c) + c^2a(c-a) \geq 0.$$ Now, I am supposed to solve this inequality by applying only the Rearrangement ...
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What inequalities for convex sets are known since the work of Scott and Awyong?

In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
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1 answer
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For $\triangle ABC$ with circumradius $R$ and nine-point center $E$, prove that $EA+EB+EC\le3R$

Given a triangle $ABC$ with the circumradius $R$, the centroid $G$, and the nine-point center $E$. Prove that $$EA+EB+EC\le 3R$$ Note. I'm trying to use the vector equation below to prove this ...
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Side lengths of a triangle satisfy $ab+bc+ca=3$. Prove that $3≤a+b+c≤2√3$ [duplicate]

The lengths $a,b$ and $c$ are the side lengths of a triangle that satisfy $ab+bc+ca=3$. Prove that $3≤a+b+c≤2√3$ I was able to find the minimum bound through some simple algebra: $a+b+c=\sqrt{a^2+b^2+...
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2 answers
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If $H$ is the orthocenter and $D,E,F$ are foot of altitudes prove that $\frac{AD}{HD}+\frac{BC}{HE}+\frac{CF}{HF}≥9$

Let ABC be a triangle with $H$ as its the orthocenter and $AD,BE,CF$ as the three altitudes. Prove that $$\frac{AD}{HD}+\frac{BC}{HE}+\frac{CF}{HF}≥9$$ I substituted $AD$ for $AH+HD$, changing the ...
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2 votes
1 answer
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If $2$ altitudes of a triangle are $9$ and $40$ then find the minimum perimeter

If $2$ altitudes of a triangle with integer side lengths are $9$ and $40$ units in length, then find the minimum possible perimeter of the triangle Since the altitude is the shortest distance from a ...
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1 vote
1 answer
61 views

proving $\min(OR,PR,QR)+OR+PR+QR<OQ+QP+PO$

In the figure prove that $$\min(OR,PR,QR)+OR+PR+QR<OQ+QP+PO$$ This is a stronger form of the inequality $OR+PR+QR<OQ+QP+PO$ that can be easily proven.I have made some constructions in the ...
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Equivalence of statements via geometric inequality on $4$ Points

Let us consider $4$ Points $A,B,C,D$ on a line $g$. We are to prove: If $\vert AP\vert+\vert PD\vert >\vert BP\vert+\vert CP\vert $ for all $P$ not on $g$, then $\vert AB\vert=\vert CD\vert$. I ...
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1 vote
2 answers
114 views

Prove this geometric inequality

Given a triangle $\triangle ABC$, let $D$ and $E$ be on points $BC$ such that $BD=DE=EC$.The line $p$ intersects $AB,AD,AE,AC$ at $K,L,M,N$ respectively. Prove that $KN ≥ 3LM$ My attempt: I think ...
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3 answers
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In which condition,the triangle have the maximum triangle area?

These trangles have the same perimeter 2q,in which condition,can we have the trangle with the maximum area? I have tried to use the Heron's formula. $S=\sqrt{q(q-a)(q-b)(q-c)}$ I have sought the ...
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