0
votes
1answer
24 views

Reverse triangle inequalities with three elements

Could you help me to show that $$ |a-b-c|\geq |b|-|a|-|c| $$ ?
2
votes
1answer
60 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, ...
0
votes
3answers
43 views

Triangle inequality and its equality

How do I prove this? $$|x+y|=|x|+|y|\Leftrightarrow xy\geq0$$ I tried to use the triangle inequality, but I didn't get so far... Thanks!
4
votes
1answer
178 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 ...
1
vote
0answers
40 views

Howto prove that $\sum_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$

let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that ...
5
votes
3answers
111 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 ...
2
votes
2answers
97 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 $$CP > \frac 1 2 (AC+BC-AB)?$$
2
votes
1answer
81 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
101 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 ...
5
votes
1answer
85 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
2
votes
1answer
87 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? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
0
votes
0answers
32 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
0answers
58 views

Cube Euclidean Metric Triangle Inequality?

I'm trying to prove that $d((x_1,y_1),(x_2,y_2)) = \sqrt[3]{|x_2 - x_1|^3 + |y_2 - y_1|^3}$ defines a metric. The problem is that I only know how to prove the triangle inequality for the Euclidean ...
3
votes
4answers
69 views

Limit on the expression containing sides of a triangle

To find the bounds of the expression $\frac{(a+b+c)^2}{ab+bc+ca}$, when a ,b, c are the sides of the triangle. I could disintegrate the given expression as $$\dfrac{a^2+b^2+c^2}{ab+bc+ca} + 2$$ and ...
1
vote
1answer
191 views

Zero “norm” properties

I have seen the claim that the l0-norm ($\|X\|_0$ = support(X)) is a pseudo-norm because it does not satisfy all properties of a norm. I thought it to be triangle inequality, but am not able to show ...
7
votes
1answer
530 views

Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
1
vote
1answer
111 views

Triangle optimization problem

Let $a,b,c$ be the sides of a triangle , then what is the maximum and minimum values (if exist) of the following quantities (i) $\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$ (ii) ...
0
votes
2answers
18 views

Triangle inequlity improvment with the angle conditions

I was working on how to proof $a+b \leq x+y+z$? Apply triangle inequity to the triangle ADC, $x+z \geq a$ Apply triangle inequity to the triangle DCB, $y+b \geq z$ Adding above inequities, ...
1
vote
2answers
121 views

Triangle $\Delta ABC$ , $a,b,c$ are in G.P.

If in a triangle $\Delta ABC$ the sides $a,b,c $ are in Geometric Progression.Find out the range of common ratio of the Geometric Progression. I understood that the twist is that we are bound under ...
1
vote
1answer
106 views

How to prove triangle inequality for given formula?

How to prove that given formula $\frac{(P-Q)^2}{P}+\frac{(P-Q)^2}{Q}$ satisfies triangle inequality ?
7
votes
1answer
98 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
2
votes
3answers
811 views

Proof of Cauchy–Schwarz inequality

I was reading about the Cauchy–Schwarz inequality from Courant, Hilbert - Methods Of Mathematical Physics Vol 1 and I can not understand what they mean when they said the line that has been ...
2
votes
3answers
155 views

Sides of triangle and an altitude

Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Let $h$ be the altitude drawn on the side of length $a$ Then is $a^2 + 4h^2 - (b+c)^2$ always negative ?
-2
votes
3answers
2k views

Triangle inequality for subtraction?

Is the following inequality(that looks like the triangle inequality) valid: $|a - b| \leq |a| - |b|$ Why?
3
votes
2answers
136 views

Showing that $ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$

I would like to show that: $$ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$$ where $\alpha, \beta, \gamma$ are the angles of a triangle. I know that the inequality $$ ...
3
votes
1answer
359 views

Is this a norm? (triangle inequality for weighted maximum norm)

I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated. Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ ...
2
votes
1answer
172 views

Trigonometric inequality for angles in triangle

Let $A, B, C$ be angles in a triangle. Is the following inequality $$4\cos A \le 1 + \cos\left(\frac{B-C}{2}\right)$$ true? I just assume it but don't have a proof. Thank you for your help.
1
vote
4answers
175 views

Inverse triangle equality [duplicate]

Possible Duplicate: Why exactly can you take the absolute value of one side of this inequality and assume it is still true? Why is $||a|-|b|| \ge |a|-|b|$, tried a lot (like comparing to ...