# Tagged Questions

22 views

### $S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
43 views

### 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 ...
228 views

### 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 ...
42 views

82 views

### Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
222 views

114 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 ...