How can i prove that $x\leq \tan(x)$ for any $x$ in $[0,\frac{\pi}{2})?$

In the $\Delta ABC$, let $r_a,r_b,r_c$ be the exradii, $S$ the area of the triangle, $a,b,c$ its sides, $p$ the semiperimeter and $r$ the inradius. Show that the following inequality holds: $$r_ar_b+r_br_c+r_cr_a \ge 2\sqrt3 S+\frac {abc}{p}+r^2.$$ I tried to express the exradii in terms of $p,a,b,c$, but I didn’t get to show that the inequality […]

Given $a,b,c$ are the sides of a triangle. Prove that $\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}<2$. My attempt: I could solve it by using the semiperimeter concept. I tried to transform this equation since it is a homogeneous equation as $f(a,b,c)=f(ta,tb,tc)$. I considered $a+b+c=1$ and thus the inequality reduces to $$\dfrac{a}{1-a}+\dfrac{b}{1-b}+\dfrac{c}{1-c}<2$$. This is getting quite difficult to prove, as I […]

For a triangle $ABC$, let $m_{a}$, $h_{a}$ be $A$-median, $A$-altitude. Define $m_{b}$,$h_{b}$ and $m_{c}$,$h_{c}$ likewise. Prove that $\dfrac{h_{a}}{m_{b}}+\dfrac{h_{b}}{m_{c}}+\dfrac{h_{c}}{m_{a}}\leq 3$ I have no solution.

Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that: $$8r+2R\le AM_1+BM_2+CM_3\le 6R$$ The second part is easy, since each of $AM_1$ is a chord, $AM_1\le 2R$, so $\sum AM_1 \le 6R$. But the […]

When I was drawing some points on paper and studied the distances between them, I found that an inequality holds for many sets of points. Suppose that we have $2$ blue points $b_1,b_2$ and $2$ red points $r_1,r_2$ in the Euclidean plane. Then using the triangular inequality, it is easy to see that the following […]

As I studying geometric inequalities, one of those famous inequalities is $$a^2+b^2+c^2\le 9R^2$$ I did some research and I found that there is a proof (not exactly the this inequality but an useful identity) of this on geometry revisited book section 1.7. the identity is $$OH^2=9R^2-(a^2+b^2+c^2)$$ where $H$ is orthocenter and $O$ is circumcenter. the […]

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China The Olympic book, but it doesn’t show a solution in book, it says it is a “W.Blaschke 1917” result, but I can’t find it. Can you […]

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?

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