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]
446 questions
2
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1 answer
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 can't solve it.
Any help appreciated.
abc...
- 4,904
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Inequality concerning segments of an interior point to vertices of a triangle
Let $ABC$ be a triangle and it has an interior point $P$ inside. Show that: $$BC\cdot BP\cdot CP + CA\cdot CP\cdot AP + AB\cdot AP\cdot BP\geq AB\cdot BC\cdot CA$$
If can, how?
Or does the triangle need specific conditions to match such…
Luis Lee
- 11
1
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1 answer
Solving an inequality fot hyperbolas
I am trying to solve:
$$\frac{1}{x-2} < \frac{1}{x+2}. $$
I think I have done so by sketching the graph,
namely $-2 < x < 2 $ but I would like to see how I could have done this algebraically.
TIA
Simon
- 11
1
vote
1 answer
Geometric Inequality $(a+b)(b+c)(c+a)(s-a)(s-b)(s-c)\leq (abc)^{2}$
everyone.
$a$,$b$,$c$ are three sides of a triangle.
Prove or disprove the following.
$(a+b)(b+c)(c+a)(a+b-c)(b+c-a)(c+a-b)\leq 8(abc)^{2}$
I know two inequalities.
$8(s-a)(s-b)(s-c)\leq abc~$ , $~(a+b)(b+c)(c+a)\geq 8abc$
But for the…
chloe_shi
- 2,855
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1 answer
If $m_a$ be the length of the median on side $a$ of acute angled triangle $ABC$ then $(m_a)^2 \le 2b^2c^2(1+\cos A)$..
Is this following inequality true?
If $m_a$ be the length of the median on side $a$ of acute angled triangle $ABC$ then $(m_a)^2 \le 2b^2c^2(1+\cos A)$..
After simplifying I am getting $ \frac{1}{bc} < \frac{4((b+c)^2-a^2)}{2b^2+2c^2-a^2}$ .But what…
user321656
0
votes
1 answer
Let P be any point inside a triangle $\triangle ABC$. Geometric Inequality
Let P be any point inside a triangle $\triangle ABC$. $A$ and $P$ are joined and extended so that $AP$ when extended intersects $BC$ on $D$. Similarly define $E$ and $F$ on $CA$ and $AB$. Prove that $PD + PE + PF < \max(AB , BC , CA)$.
user321656