This is a tag for geometric problems involving inequalities.
This tag is named Geometric Inequalities. Geometric problems with inequalities belong here. This tag also includes trigonometric problems with inequalities.
Questions tagged [geometric-inequalities]
460 questions
39
votes
1 answer
How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$
Zhautykov Olympiad 2015 problem 6 This links discusses the olympiad problem which none of students could solve , meaning it is very hard.
Question:
The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$,…
math110
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How to prove that $ \sin \angle{GAB}+\sin \angle{GBC}+\sin \angle{GCA} \le \frac{3}{2} $ for a triangle $ABC$ with centroid $G$?
Let $ G $ be the centroid of $ \triangle ABC $ , such that $ \measuredangle{GAB}=x,\measuredangle{GBC}=y,\measuredangle{GCA}=z $.
How do I prove that :
$$ \sin x +\sin y +\sin z\le \frac{3}{2} $$
piteer
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20
votes
3 answers
The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$
Given a triangle $ABC$, and $M$ is an interior point. Prove that:
$\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$.
When does equality hold?
user252487
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19
votes
4 answers
Proving $a^2x+b^2y+c^2z\geq 4[ABC]\sqrt{xy+xz+yz}$, for $a$, $b$, $c$ the sides of $\triangle ABC$, $[ABC]$ its area, and $x$, $y$, $z$ positive reals
I did a proof for the inequality below, and I would like know if anyone also has a trigonometric proof for this inequality.If you have a trigonometric demonstration, please post your solution. This problem appeared in the American Mathematical…
18
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1 answer
Geometric inequality $\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$
Let $P$ be a point inside $\triangle{ABC}$.
Let $AP=R_a$, $BP=R_b$, $CP=R_c$, $AB=c$, $BC=a$ and $CA=b$. Prove that:
$$\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$$
I tried to use C-S, but without success because the…
Michael Rozenberg
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16
votes
6 answers
Unit square inside triangle.
Some time ago I saw this beautiful problem and think it is worth to post it:
Let $S$ be the area of triangle that covers a square of side $1$. Prove that $S \ge 2$.
hOff
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15
votes
3 answers
A curious triangle inequality
Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that
\begin{equation}
|PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|.
\end{equation}
ecstasyofgold
- 884
14
votes
1 answer
How prove $\sum\limits_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum\limits_{cyc}h_{a}}$
Question:
Consider a triangle $\Delta ABC$ with altitudes $h_{a}$, $h_{b}$ and $h_{c}$, where $AB=c$, $BC=a$ and $AC=b$.
Show that for any $P$
$$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$
My try: the inequality is…
math110
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4 answers
Given triangle side lengths $a, b, c$, show that $3\left(a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\right)\geq b\left(a+b-c\right)\left(a-c\right)\left(c-b\right)$
If you're interested in IMO 1983-style inequalities, please consider the following problem:
Given three positive real numbers $a, b, c$ that form the side lengths of a triangle, prove inequality:
$$3\left ( a^{2}b\left ( a- b \right )+ b^{2}c\left (…
user685500
13
votes
8 answers
Extreme of $\cos A\cos B\cos C$ in a triangle without calculus.
If $A,B,C$ are angles of a triangle, find the extreme value of $\cos A\cos B\cos C$.
I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps.
On this topic we learned also about Cauchy inequality, but…
Den
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12
votes
3 answers
An inequality about the sum of distances between points : same color $\le$ different colors?
When I was drawing some points on paper and studied the distances between them, I found that an inequality holds for many sets of points.
Suppose that we have $2$ blue points $b_1,b_2$ and $2$ red points $r_1,r_2$ in the Euclidean plane. Then using…
mathlove
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11
votes
1 answer
Showing that $\sin^2x\cdot\sin^22x\cdot\sin^24x\cdot\sin^28x\cdots\sin^22^nx\leq\frac{3^n}{4^n}$
Show that $$\sin^2x\cdot\sin^22x\cdot\sin^24x\cdot\sin^28x\cdots\sin^22^nx\leq\frac{3^n}{4^n}$$
I understand the result of an arithmetic sequence $(\sin1^\circ)(\sin3^\circ)(\sin5^\circ)…(\sin89^\circ)$, how about the geometric sequence case?
Ziyi Guo
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11
votes
3 answers
Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.
Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest.
I tried using Ptolemy's theorem but don't know how to make use of inequalities it gives.
I'd be really…
SamHar
- 173
11
votes
2 answers
How to prove this geometry inequality (1) with $2(DF+EF)\ge BC$
There is a picture which hope to illustrate the configuration:
$\triangle ABC$ is such that $\angle A=\dfrac{2\pi}{3}$, and $F$ is the midpoint of $BC$, and $D,E$ lie on $AB,AC$ respectively, such that $DE ||BC$.
Show that:
$$2(DF+EF)\ge BC$$
maybe…
math110
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10
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2 answers
A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$
Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that:
$$8r+2R\le AM_1+BM_2+CM_3\le 6R$$
The second part is easy, since each of…
Sawarnik
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