Questions tagged [geometric-inequalities]
This is a tag for geometric problems involving inequalities.
0
votes
3answers
48 views
Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.
How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$.
What is the geometric intuition behind this inequality?
1
vote
1answer
32 views
Geometric inequality
Given Triangle ABC, let D be mid point of BC and let E and F be two point on sides AB and AC respectively such that angle EDF is right angle (90°). Prove that BE + CF > EF. I was able to do it by ...
1
vote
1answer
53 views
Help in solving this geometric inequality
I have the following geometry problem with me:
"The altitudes through the vertices A,B,C of an acute angles triangle meet the opposite sides at D,E and F respectively, and AB > AC. The line EF meets ...
0
votes
1answer
53 views
Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$
For $a_1, \ldots , a_n \in \mathbb{R}, a_1 < a_2 < \cdots <a_n$ and $a_i \ne 0$, show that
$\dfrac{n}{a_1 - a_0} + \dfrac{n - 1}{a_2 - a_1} + \cdots + \dfrac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^...
1
vote
0answers
45 views
Norm inequality problem on a finite dimensional vector space.
My question is how do we prove that last inequality?
3
votes
1answer
111 views
For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$
In the interior of $\triangle ABC$ we take the arbitrary point $P$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\...
0
votes
1answer
82 views
Maximizing $\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$ for $x$, $y$, $z$ the angles of a non-acute triangle
$x$, $y$, $z$ are the angles of a triangle, one of which is not less than $\frac{\pi}{2}$. Find the maximum value of the expression $$\cos{x}+\cos{y}+\cos{z}+\cos{(x-y)}+\cos{(y-z)}+\cos{(z-x)}$$
I ...
1
vote
1answer
43 views
A triangle has sides x, x+4, and 3x−5. What is the possible range of x?
A triangle has sides $x, x+4,$ $and$ $3x−5$. What is the possible range of x?
The answer to this is not:
$2x-9$ $<$ $x$ $<$ $4x-1$
Which is gotten from:
--> $(3x-5)-(x+4)<x<3x-5+x+4$
...
7
votes
1answer
187 views
A nice IMO 1983 inequality from a stronger inequality
If you are interested in IMO $1983$ please see:
$$3[a^2b(a-b)+b^2c(b-c)+c^2a(c-a)]\geqq b(a+b-c)(a-c)(c-b),$$
where $a,b,c$ are three side-lengths of a triangle.
If $c≠{\rm mid}\{a,b,c\}$, the ...
2
votes
1answer
50 views
Prove that in every triangle the inequality $a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $ takes place
Prove that in every triangle the inequality $$a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $$ takes place, with the usual notations ($a,b,c$ lengths of sides, $r_a, r_b, r_c$ radii of coresponding ...
4
votes
1answer
79 views
Proving the inequality $\angle A+\angle COP < 90^\circ$ in $\triangle ABC$
In an acute angled $\triangle ABC$, $AP \perp BC$and $O$ is its circumcenter. If $\angle C \ge \angle B + 30^\circ$, then prove that $$\angle A + \angle COP < 90^\circ$$
My Attempt:
Extending the ...
1
vote
1answer
50 views
I cannot reproduce a geometric result algebraically
Take a parabola $u(x)=ax^2 + bx + c$, where $a<0$, and draw a secant on an upward-sloping portion of the parabola from $(w-h, u(w-h))$ to $(w+h, u(w+h))$, for some $w \leq -\frac b{2a} - |h|$. Now ...
1
vote
1answer
27 views
Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $
Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $
I feel like the products between different variables (i.e. not x^2, y^2, z^2) give this inequality the $ \gt $ sign and I don't ...
3
votes
1answer
46 views
Equality in triangle obtuse
$m_1, m_2, m_3 $ are sides-lengths of a triangle such that $m_1\sqrt{m_1}+m_2\sqrt{m_2}=m_3\sqrt{m_3}$.
Prove that this triangle is an obtuse-angled triangle.
I don't have idea make run this ...
9
votes
1answer
399 views
Patterns in inequalities of triangle involving angles.
I was reading this page and wondered as why, inequalities for $\cos A$ (with argument $A$) become the same inequality for $\sin\frac{A}{2}$ (with argument $\frac{A}{2}$), similarly for $\tan$ and $\...
10
votes
3answers
381 views
Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.
Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest.
I tried using Ptolemy's theorem but don't know how to make ...
0
votes
1answer
38 views
Inequality triangle Radon substitutions
I have this inequality:
$$\sum \frac {a^3}{p-a}\geq 8(2R-r)^2$$
I have tried using Radon substitutions and I get this:
$$\sum \frac{(y+x)^3}{x}\geq 8(2R-r)^2$$
I know from Holder that :
$$\sum \frac{(...
2
votes
2answers
59 views
Is there a right triangle with angles $A$, $B$, $C$ such that $A^2+B^2=C^2$?
A right angle triangle with vertices $A,B,C$ ($C$ is the right angle), and the sides opposite to the vertices are $a,b,c$, respectively.
We know that this triangle (and any right angle triangle) has ...
1
vote
1answer
147 views
Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$
Let $a,b,c\in \Bbb R^+$ such that $a+b+c=abc$. Prove that $$\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$$
Idea 1.From $a+b+c=abc\Leftrightarrow \frac{1}{ab}+...
4
votes
0answers
53 views
Generalizing Heron's Formula for Cyclic $n$-gons
Consider the following extension of Heron's Formula.
For a cyclic $n$-gon $C$ with side lengths $x_1, x_2, \dots, x_n$ and semi-perimeter $P = \frac{1}{2} \left( x_1 + x_2 + \dots + x_n\right)$ define:...
3
votes
2answers
112 views
Write in a concise form a set in 4-dimensional space.
I have the following subset of the 4-dimensional space $(x_1,x_2,x_3,y)$:
\begin{align}
\mathcal I=&\left\{x_1=0\, , \, y\in[0, 1]\, , \, 0 \leq x_2\leq y\, , \, 0 \leq x_3\leq y\right\}\cup\\ &...
5
votes
1answer
101 views
A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$
Let $\triangle ABC$ have sides $BC=a$, $CA=b$, and $AB=c$. Let $m_a$, $m_b$, $m_c$ be the medians to $BC$, $CA$, and $AB$, respectively. Prove that
$$(ab+bc+ca)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{...
0
votes
2answers
51 views
A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$? [duplicate]
Let $a_1,\dots,a_d,b_1,\dots,b_d$ be positive real numbers. Then
$$ \prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$$
and equality holds if and only if $\frac{a_i}{a_j}=\frac{...
-1
votes
3answers
69 views
geometry inequality
$M$ is a point in $\triangle ABC$. $AM$ intersect with $BC$ at $A_1$. $BM$ intersect with $AC$ at $B_1$. $CM$ intersect with $AB$ at $C_1$.
Proof that: $$AA_1 \times BB_1 \times CC_1 \geq 27 (MA_1 \...
2
votes
2answers
72 views
Proof using AM-GM inequality
The questions has two parts:
Prove
(i) $ xy^{3} \leq \frac{1}{4}x^{4} + \frac{3}{4}y^{4} $
and
(ii) $ xy^{3} + x^{3}y \leq x^{4} + y^{4}$.
Now then, I went about putting both sides of $\sqrt{xy}...
1
vote
2answers
44 views
Find at least one solution to matrix inequality
I have the following problem posed:
find at least one vector $\mathbf{x}$ such that
$$
A\mathbf{x} + \mathbf{b} \geqslant \mathbf{0}
$$
for a given matrix $A$ and vector $\mathbf{b}$. Nothing is ...
3
votes
3answers
224 views
Difficult inequality with three real variables
For any real $e, t, \sigma$ such that
\begin{aligned}
\label{s}
0&<e<1\,,\\
0&<t<\pi\,,\qquad\qquad\qquad(1)\\
-\pi/2&\leqslant\sigma\leqslant\pi/2
\end{aligned}
the ...
0
votes
2answers
71 views
Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [closed]
A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
1
vote
5answers
658 views
Minimizing $\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$
What is the minimum value of
$$\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$$
I tried getting the first derivative, but I can't solve the equation when I put $y = 0$.
Methods without using calculus ...
0
votes
2answers
87 views
INMO 1998 question
This was question 5 of 1998 INMO.
Suppose $a,b,c$ are three real numbers such that the quadratic equation $$ x^2 - (a +b +c )x + (ab +bc +ca) = 0 $$has roots of the form $\alpha + i \beta$ where $\...
1
vote
1answer
65 views
Complex number inequalities
I got the following question below :
I tried and got this as the region , is this correct ?
0
votes
0answers
38 views
lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.
Let $ABC$ be an acute triangle.
The goal is to prove:
\begin{align*}
&(a)\;\;\;\pi(2R-r) < aA+ bB + cC < 4(2R-r)\\[2pt]
&\qquad\qquad\text{[where in the above, $A,B,C$ denote the ...
2
votes
1answer
70 views
Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$
If $a,b,c$ are three sides of a triangle then I need to prove that
$$
{\frac {a}{a+b}}+{\frac {b}{b+c}}+{\frac {c}{c+a}}\geq \frac{3}{2}\, \left( {
\frac {a}{a+b}}+{\frac {b}{b+c}} \right) \left( {\...
2
votes
1answer
89 views
Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$
Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$
My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is ...
3
votes
2answers
66 views
Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$
I am dealing with the test of the OBM (Brasilian Math Olimpyad), University level, 2017, phase 2.
As I've said at others topics (questions 1 and 2, this last yet open, here), I hope someone can help ...
0
votes
5answers
110 views
Finding the values of $x$ and $y$ for which $x^2 + y^2$ is a minimum [closed]
Three squares are shown in the diagram.
The largest has side $AB$ of length $1$. The
others have side $AC$ of length $x$, and side
$DE$ of length $y$. As $D$ moves along $AB$,
the values of $x$ and $...
2
votes
1answer
31 views
$A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$
Let $ABCD$ a square of length $ 2$.
If we consider $A_1, A_2\in [AB]$ s.t. $AA_1 < AA_2 < 1$ and
$B_1, B_2\in [BC]$ s.t. $BB_1 < BB_2< 1$ and $C_1, C_2\in [CD]$
$D_1, D_2\in [DA]$ with ...
-1
votes
1answer
87 views
Inequality about Schur
Let a,b,c be the sides of triangle such that $a+b+c=1$. Prove that $$5(ab+bc+ca)\geq18abc+a+b+c$$
I tried to prove:
$$5(ab+bc+ca)\geq18abc+a+b+c$$$$10(ab+ac+bc)\geq36abc+2(a+b+c)$$$$a(5b+5c-2-12bc)+b(...
4
votes
1answer
221 views
inequality in geometry
Given triangle $ABC$ has $BC=a$, $CA=b$, $AB=c$ and $M$ is a point of the triangle plane. Prove that:
\begin{align*}
\cos \dfrac{A}{2} \cdot MA+\cos \dfrac{B}{2} \cdot MB+\cos \dfrac{C}{2} \cdot ...
1
vote
0answers
90 views
Known proofs of isodiametric inequality
Isodiametric inequality is the following statement:
Suppose that $A\subset\mathbb R^n$ is a compact set of diameter at most 2. Then the volume of $A$ is less than or equal to the volume of a unit ...
1
vote
2answers
290 views
Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$
Prove that if $a,b,c$ are the lengths of the edges of a given triangle, then the following inequality holds:
$\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + ...
2
votes
3answers
67 views
Prove inequality inside a triangle
"Let $ABC$ be a triangle with centroid $G$. A line $PQ$ is drawed in the triangle such that it passes through $G$ and intersects the sides $AB$ and $AC$ in $P$ and $Q$ respectively."
Prove that:
$$\...
1
vote
6answers
127 views
Proving $\frac{1}{\sin(A/2)}+\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\ge 6$, where $A$, $B$, $C$ are angles of a triangle [duplicate]
If $A$, $B$, and $C$ are the angles of a triangle, then $$\frac{1}{\sin \left(\frac{A}{2}\right)}+\frac{1}{\sin\left(\frac{B}{2}\right)}+\frac{1}{\sin\left(\frac{C}{2}\right)}\ge 6$$
I have used ...
0
votes
1answer
50 views
inequality proving for intersection
AB and CD are line segments, $AB=CD=1$, intersecting in point O, $\enspace$ $AB\cap CD =O$, $\angle AOC=60^{\circ}$. Prove that $AC+BD\geq1$. $\enspace$
What I tried:
$AO+BO=1$, $\enspace$ $CO+DO=1$ ...
-1
votes
3answers
60 views
Draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$
I need to draw a graph for the inequality $(x_1 - \frac{1}{2})^2 + (y_1 - \frac{1}{2})^2 < (x_2 - \frac{1}{2})^2 + (y_2 - \frac{1}{2})^2$, where $x_1 \in [0, \frac{1}{2}]$, $x_2 \in [\frac{1}{2}, 1]...
0
votes
1answer
91 views
Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.
Problem
For any $\triangle ABC$, let $a,b,c$ be the length of the side respectively. Show that $$(2b+2c-2a)^3(a+b+c)\geq18a^2bc.$$
1
vote
2answers
66 views
Minimal area of a right triangle with inradius $1$
I got this question, solved it, then forgot how I solved it.
What is the minimal area of a right triangle with inradius $1$?
My attempt:
$r=\frac{a+b-c}2$, so $a+b=c+2$
$a^2+b^2=c^2$
This gives ...
2
votes
2answers
77 views
A geometric inequality for a triangle ABC
I have to prove that:
$ \frac {a^2}{w_a^2} + \frac {b^2}{ w_b^2} +\frac {c^2}{ w_c^2} \ge 4,$
for the sides $a,b,c$ of the triangle $ABC,$ $ w_a, w_b, w_c $ the angle bisectors and $s$ its ...
3
votes
2answers
102 views
Proving a weird inequality
Recently I saw this interesting inequality problem from Singapore Math Olympiad 2016 (Open Section, Special Round), but I am not sure how to start, here it goes:
Real numbers $a,b,c$ are such that $...
2
votes
1answer
54 views
A geometrical inequality with a sphere
A bee flied 4 meters (in total) and back to the original spot. Prove its path can be inscribed in a sphere with radius 1m.
I am bad at $3D$ geometry, so I tried to reduce it to a plane, but I still ...
