Newbetuts

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$

How to prove this geometry inequality (1) with $2(DF+EF)\ge BC$

How to show an inequality in an inner product space?

Minimize area of n-gon circumscribed around unit circle

Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $

How to prove this inequality involving $\tanh$?

If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$.

Oppenheim's Inequality for triangles, American Mathematical Monthly problems

Showing that $\sin^2x\cdot\sin^22x\cdot\sin^24x\cdot\sin^28x\cdots\sin^22^nx\leq\frac{3^n}{4^n}$

How prove $\sum\limits_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum\limits_{cyc}h_{a}}$

A curious triangle inequality

Prove that: $ m_a + m_b + m_c \le 4R + r $

Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

Unit square inside triangle. [duplicate]

Alternate proof for $a^2+b^2+c^2\le 9R^2$

New posts in geometric-inequalities

For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$

A bound for $\sqrt\frac{b+c-a}a+\sqrt\frac{c+a-b}b+\sqrt\frac{a+b-c}c$ in a triangle

Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$

How to prove this geometry inequality (1) with $2(DF+EF)\ge BC$

How to show an inequality in an inner product space?

Minimize area of n-gon circumscribed around unit circle

Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $

How to prove this inequality involving $\tanh$?

If $a$, $b$, and $c$ are sides of a triangle, then $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$.

Oppenheim's Inequality for triangles, American Mathematical Monthly problems

Showing that $\sin^2x\cdot\sin^22x\cdot\sin^24x\cdot\sin^28x\cdots\sin^22^nx\leq\frac{3^n}{4^n}$

How prove $\sum\limits_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum\limits_{cyc}h_{a}}$

A curious triangle inequality

Prove that: $ m_a + m_b + m_c \le 4R + r $

Permutation of points $P_i\in X$ such that $\sum^n_{j=1}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8$

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

Unit square inside triangle. [duplicate]

Alternate proof for $a^2+b^2+c^2\le 9R^2$