2
votes
0answers
42 views

How prove $\sum_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum_{cyc}h_{a}}$

Question: let $\Delta ABC$,and the altitude is $h_{a},h_{b},h_{c}$,where $AB=c,BC=a,AC=b$ and for any $P$ show that $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
0
votes
1answer
35 views

Geometric Application of Cauchy-Schwarz Inequality Problem

I have been struggling with this problem, and would like to prove the inequality using the Cauchy-Schwarz Inequality: The vertices of a fixed triangle are $A$,$B$ and $C$, and $P$,$Q$ and $R$ lie on ...
2
votes
1answer
43 views

Area of a triangle whose each side is less than 2 and greater than1.

What is the area of a triangle if each of its sides is greater than 1 and less than 2? My Try:Let a,b,c be the sides of triangle,then ...
1
vote
1answer
42 views

An angular inequality

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on ...
11
votes
2answers
229 views

Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$ \text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)), $$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
1
vote
1answer
19 views

Inequality involving two altitudes of an isosceles triangle and its base

I am trying to solve the following multiple choice problem: $ABC$ is a triangle such that $AB=AC$. Let $D$ be the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from ...
1
vote
2answers
40 views

Relation between the GM of two sides of a triangle and the bisector of angle between them

I am trying to solve the following multiple choice problem : Let the bisector of the angle $C$ of a triangle $ABC$ intersect the side $AB$ at a point $D$. Then the geometric mean of $CA$ and $CB$ ...
5
votes
3answers
222 views

Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle

I just got a long expression: $$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$ and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides I really need tips how to ...
2
votes
1answer
57 views

How prove this Pedoe inequality $a^2_{1}(b^2_{2}+c^2_{2}+d^2_{2}-a^2_{2})+b^2_{1}(a^2_{2}+c^2_{2}+d^2_{2}-b^2_{2})+\cdots\ge 16F_{1}F_{2}$

Question: let $A_{i}B_{i}C_{i}D_{i}(i=1,2)$ be two quadrilaterals,and let $$B_{i}C_{i}=a_{i},C_{i}D_{i}=b_{i},D_{i}A_{i}=c_{i},A_{i}B_{i}=d_{i},i=1,2$$ and let $F_{i}$ denote the areas of ...
4
votes
1answer
54 views

Proof of Ptolemy's inequality?

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge ...
4
votes
1answer
43 views

$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$ in a triangle?

How can I prove that ( $\small{\sum}$ denotes cyclic sum here), for any triangle $ABC$: $$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$$ I don't see where to begin even. Any hints would be ...
3
votes
1answer
58 views

How to prove $\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$?

For any triangle $ABC$, prove that: $$\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$$ I have tried many approaches but none seems to work. I noted that $\cos(\frac{B-C}2)=\frac{AM}{2R}$, where $M$ is ...
2
votes
1answer
17 views

Inequality relating to product of sides of convex quadrilateral

We have been that length of both the diagonals are equal to $x$. What can be the maximum value of the product of length of sides? It is obvious that an upper bound exists, but I can't get the ...
4
votes
0answers
25 views

An inequality concerning the sides of convex quadrilateral [duplicate]

$ABCD$ is a convex quadrilateral such that $ \angle ABC\ge120^{o} , \angle BCD\ge120^{o}$ , then is it true that $AC+BD>AB+BC+CD$ ?
3
votes
1answer
77 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
10
votes
2answers
202 views

A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$

Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that: $$8r+2R\le ...
3
votes
1answer
106 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
2
votes
2answers
31 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
4
votes
2answers
128 views

How find the $AP+\frac{1}{2}BP$ minmum value

An equilateral triangle $ABC$ such $$AB=BC=AC=2a>0$$ A circle $O$ is inscribed in triangle $ABC$,and the point $P$ on the circle $O$. Find the minimum $$AP+\dfrac{1}{2}BP$$ My idea: let ...
0
votes
1answer
28 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
1
vote
1answer
25 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
5
votes
1answer
44 views

Proof of the Barrow's Inequality?

Barrow's inequality states that if $P$ is any point inside triangle $ABC$, and $PU$, $PW$, and $PV$ are the angle bisectors, then the following inequality holds, $PA+PB+PC\geq 2(PU+PV+PW)$. I know ...
2
votes
1answer
84 views

What is the converse of the triangle inequality?

It's usual when presenting a theorem to also present its converse. Surprisingly, I've never seen the triangle inequality's converse stated. Triangle inequality: If the sides of a triangle are a, b, ...
1
vote
0answers
93 views

product of sums

This is a question which has puzzled me for a while. I would be very thankful if somebody can help me with it. Assume you have $S$ rectangles appearing in front of your screen one by one. Each ...
0
votes
0answers
81 views

Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
4
votes
1answer
214 views

Inequality in triangle involving side lenghs, medians and area

A, B and C are the vertices of a triangle. Denote $m_a$, $m_b$ and $m_c$ the medians from A, B and C. Prove the inequality: $$\sum_{cyc}{a^2bcm_a}\geq\sum_{cyc}{cS(a^2+b^2)}$$where a, b and c are the ...
5
votes
3answers
145 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
6
votes
0answers
160 views

Solution set of inequalities in $\mathbb{R}^6$

Let $\theta\in (0,1)$ fixed. We define $A_1$ be the set of all $(a_1,a_2,a_3,a_4,a_5,a_6)\in (0,1)^6$ such that the following conditions hold: \begin{equation} (1) \quad a_2\le a_1, a_4\le a_3, a_6 ...
2
votes
2answers
102 views

How do I prove that $CP > \frac 1 2 (AC+BC-AB)$? [closed]

Given is the triangle $ABC$ with point $P$ on side $AB$. How do I prove that $$CP > \frac 1 2 (AC+BC-AB)?$$
1
vote
1answer
64 views

Triangle inequality not making sense in this problem.

I'm working through "An Introduction to Inequalities" by Bellman and Beckenbach. They're discussing the path of a reflected ray of light, and they make a statement that seems kind of un-intuitive to ...
6
votes
2answers
122 views

Unequal circles within circle with least possible radius?

It is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to ...
2
votes
1answer
86 views

Inequality in a triangle

Let $O$ be the circumcenter and $H$ the orthocenter in a triangle with sides $a, b, c$. Is it true that $$aOA^2+bOB^2+cOC^2 \ge aHA^2 + bHB^2 + cHC^2$$ or equivalently $$(a+b+c)R^2 \ge aHA^2 + bHB^2 + ...
2
votes
2answers
111 views

How prove this stronger than Weitzenbock's inequality:$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$

In $\Delta ABC$,$$AB=c,BC=a,AC=b,S_{ABC}=S$$ show that $$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$$ I know this Weitzenböck's_inequality $$a^2+b^2+c^2\ge 4\sqrt{3}S$$ But my ...
0
votes
1answer
96 views

Inequality in triangle involving medians

Let $ABC$ be a triangle and $M$ a point on $(BC)$, $N$ a point on $(CA)$ and $P$ a point on $(AB)$ such that the triangles $ABC$ and $MNP$ have the same centroid. Does the following inequality hold: ...
10
votes
2answers
264 views

Unit diameter pentagons with maximum area

In the euclidean plane, if one considers the set of quadrilaterals having unit diameter (maximum distance between two points in the convex envelope), it is quite easy to give a description of the ...
-1
votes
2answers
137 views

if a;b;c are the sides of a triangle then prove the inequality; or geometric inequality

let a;b;c are the sides of a triangle ABC. PROVE $(abc)^2$$\geq$$\frac{4\bigtriangleup}{\sqrt{3}}$ where $\bigtriangleup$ is the area of the triangle. clearly a;b;c are the sides opposite to the ...
2
votes
2answers
51 views

Prove, in a $\Delta ABC$ with medians $BE, CF$, $BE + CF > BC\cdot\frac32$

Consider a $\Delta ABC$ with medians $BE$ and $CF$. Prove that $$BE + CF > BC\cdot\frac32$$ Consider the following inequalities, given by the triangle inequality: $$BE + CE > BC \implies BE ...
4
votes
2answers
63 views

Third-degree cosine inequality for obtuse triangle

Suppose $\triangle ABC$ is an obtuse triangle with side lengths $a=BC, b=CA, c=AB$. I want to show that $$a^3\cos A+b^3\cos B+c^3\cos C<abc.$$ My idea is to use the cosine rule. I have $\cos ...
3
votes
1answer
98 views

Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
1
vote
1answer
33 views

How find this inequality find the maximum $z_{5}$

let $z_{1},z_{2},z_{3},z_{4},z_{5}\in C$,such $$\begin{cases} |z_{1}|\le 1,|z_{2}|\le 1\\ |2z_{3}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ |2z_{4}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ ...
5
votes
4answers
282 views

How find this maximum $S_{\Delta ABC}$

in $\Delta ABC$,and $\angle ABC=60$,such that $PA=10,PB=6,PC=7$, find the maximum $S_{\Delta ABC}$. My try:let $AB=c,BC=a,AC=b$, then $$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$ then ...
1
vote
0answers
91 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
4
votes
1answer
243 views

How prove this$\frac{1}{P_{0}P_{1}}+\frac{1}{P_{0}P_{2}}+\cdots+\frac{1}{P_{0}P_{n}}<\sqrt{15n}$

Let $P_{0},P_{1},P_{2},\cdots,P_{n}$ be $n+1$ points in the plane. Let $ d=1$ denote the minimal value of all the distances between any two points. Prove that ...
0
votes
0answers
35 views

Reverse Triangle Inequality shortest proof [duplicate]

I've been trying to prove the triangle inequality but I was wondering what is the shortest way to prove that $\forall (x,y)\in \mathbb{R}^2$, $||x|−|y||\le|x−y|$ ? thanks
0
votes
1answer
36 views

Visualize the effect of adding another constraint

I have 2 eqns $$ x_1+4x_3\leq4$$ $$ x_2+4x_3\leq4 $$ $$x_1\geq0$$ $$x_2\geq x_3\geq0$$ By drawing geometrical figure I have vertices whose co-ordinates is $(0,0,0) , (4,0,0) (0,0,1) ,(0,4,0) ...
7
votes
1answer
259 views

How can one prove this geometric inequality?

Let $ABCDEF$ be a convex hexagon with area $S$. Show that $$BD\cdot(AC+CE-EA)+DF\cdot(CE+EA-AC)+FB\cdot(EA+AC-CE)\ge 2\sqrt{3}\cdot S.$$ At some point, I found this similar problem. Thank you to ...
2
votes
2answers
42 views

Are there exist $a, b >0$ such that $az_1^Tz_1 \geq b|z_1^Tz_2|$ and $bz_2^Tz_2 \geq a|z_2^Tz_1|$?

I am looking whether these inequalities can be held at the same time $$az_1^Tz_1 \geq b|z_1^Tz_2| \\ bz_2^Tz_2 \geq a|z_2^Tz_1|,$$ where $a$ and $b$ are two positive constants, $z_i \in \mathbb{R}^2, ...
5
votes
2answers
126 views

How prove this inequality $\dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b}$

in $\Delta ABC$,prove that $$\dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b}$$ where $R$ is the circumradius and $r$ is the inradius By the way.It is well konwn that Eluer inequality $$R\ge 2r$$ and it is ...
0
votes
1answer
51 views

Generalizing a statement about points in the unit square

What is the three-dimensional version of this statement: Any $n$ points in the unit square can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 ...
5
votes
3answers
252 views

Points in unit square

Let $n$ points be given in the unit square. How to prove or disprove: the points can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\| ^2 \le 4,$$ ...