# Tagged Questions

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### 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 $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $z=0$ ...
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### 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 ...
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### 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 $CB$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
let $z_{1},z_{2},z_{3},z_{4},z_{5}\in C$,such $$\begin{cases} |z_{1}|\le 1,|z_{2}|\le 1\\ |2z_{3}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ |2z_{4}-(z_{1}+z_{2})|\le|z_{1}-z_{2}|\\ ... 4answers 282 views ### How find this maximum S_{\Delta ABC} in \Delta ABC,and \angle ABC=60,such that PA=10,PB=6,PC=7, find the maximum S_{\Delta ABC}. My try:let AB=c,BC=a,AC=b, then$$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$then ... 0answers 91 views ### Is there such an example? Is there an example of a sequence of point sets \left\{ S_{n}\right\} _{n=1}^{\infty}in which S_{n} is a set of n points inside the unit triangle, such that the minimum altitude of the triangles ... 1answer 243 views ### How prove this\frac{1}{P_{0}P_{1}}+\frac{1}{P_{0}P_{2}}+\cdots+\frac{1}{P_{0}P_{n}}<\sqrt{15n} Let P_{0},P_{1},P_{2},\cdots,P_{n} be n+1 points in the plane. Let  d=1 denote the minimal value of all the distances between any two points. Prove that ... 0answers 35 views ### Reverse Triangle Inequality shortest proof [duplicate] I've been trying to prove the triangle inequality but I was wondering what is the shortest way to prove that \forall (x,y)\in \mathbb{R}^2, ||x|−|y||\le|x−y| ? thanks 1answer 36 views ### Visualize the effect of adding another constraint I have 2 eqns$$ x_1+4x_3\leq4 x_2+4x_3\leq4 x_1\geq0x_2\geq x_3\geq0$$By drawing geometrical figure I have vertices whose co-ordinates is (0,0,0) , (4,0,0) (0,0,1) ,(0,4,0) ... 1answer 259 views ### How can one prove this geometric inequality? Let ABCDEF be a convex hexagon with area S. Show that$$BD\cdot(AC+CE-EA)+DF\cdot(CE+EA-AC)+FB\cdot(EA+AC-CE)\ge 2\sqrt{3}\cdot S.$$At some point, I found this similar problem. Thank you to ... 2answers 42 views ### Are there exist a, b >0 such that az_1^Tz_1 \geq b|z_1^Tz_2| and bz_2^Tz_2 \geq a|z_2^Tz_1|? I am looking whether these inequalities can be held at the same time$$az_1^Tz_1 \geq b|z_1^Tz_2| \\ bz_2^Tz_2 \geq a|z_2^Tz_1|,$$where a and b are two positive constants, z_i \in \mathbb{R}^2, ... 2answers 126 views ### How prove this inequality \dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b} in \Delta ABC,prove that$$\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$$and it is ... 1answer 51 views ### Generalizing a statement about points in the unit square What is the three-dimensional version of this statement: Any n points in the unit square can be labeled x_1,\ldots,x_n to satisfy the inequality$$\|x_1-x_2\|^2 ...
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 $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\| ^2 \le 4,$$ ...