# Questions tagged [geometric-inequalities]

This is a tag for geometric problems involving inequalities.

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### sum of projected area and a generalization of Oppenheim's inequality

From this post: In $\mathbb R^4$, let $U$ be a 2D plane, let $\pi_1$ be the projection from $U$ onto $xy$-plane and $\pi_2$ be the projection from $U$ onto $zw$-plane, then $\det\pi_1+\det\pi_2\le1$, ...
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### Geometric inequalities using sides, radius and semi-perimeter

Prove that: $$\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}S}+1 \ge 2\cdot\frac{4R+r}{\sqrt{3}p}$$ where $a, b, c$ are the sides of the triangle, $S$ is the area, $R$ is the circumradius, $r$ is the inradius and ...
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### Sum of distances from interior point to vertices is greater than double the sum of distances to edges

Let $\triangle ABC$, $M\in Int(\triangle ABC)$. Let $MD\perp BC$, $ME\perp AC$ and $MF\perp AB$, $D\in BC$, $E\in AC$, $F\in AB$. Prove that $$MA+MB+MC\geq 2\cdot(MD+ME+MF).$$ My only idea is to use ...
• 991
1 vote
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### Given $a,b,c$ are sides of a triangle, Prove that :- $\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$

Given $a,b,c$ are sides of a triangle, Prove that :- $$\frac{1}{3} \leq \frac{a^2+b^2+c^2}{(a+b+c)^2} < \frac{1}{2}$$ What I Tried:- I was able to solve the left hand side inequality. From RMS-AM ...
• 4,254
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### Prove that if $a,b,c$ are sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$

Prove that, if $a,b,c$ are the sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$. What I Tried:- Nothing special really came in ...
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### Systems of linear homogenous inequalities: getting started

I have a number of questions of varying difficulties related to satisfying largish (as large as possible) systems of linear inequalities. I gather these aren't easy, so I'd be happy to get numerical ...
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Let $x,y \in (0,1)$, and suppose that $$x^2-2x+y^2<0.$$ How to prove that $$-x^3-xy^2+4y^2 \ge 0.$$ holds? The motivation comes from a certain geometric problem (a bit long to describe here).