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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 easy proof: by $$d^2=R(R-2r)$$ see http://mathworld.wolfram.com/EulersInequality.html

and $$ \dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b}$$ is shaper than Eluer inequality, I try solve it,But I can't work, But after I can find This is famous inequality:see http://books.google.com.hk/books?id=hXYH2OfNRdwC&pg=PA177&lpg=PA177&dq=Gaz.+Mat.+(Bucharest)+90+(1985),+65&source=bl&ots=xP0JX0PNta&sig=RdcaDooKrw-0wvdVsj4C35E5AKY&hl=zh-CN&sa=X&ei=qXgHUsOUHMWziQeT8IG4DA&ved=0CC4Q6AEwAA#v=onepage&q=Gaz.%20Mat.%20(Bucharest)%2090%20(1985)%2C%2065&f=false (5.30),But I can't find this equality solution too,Thank you someone can find it or take this inequality methods。Thank you everyone.

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$l_{a}$ is $A$-inner angle bisector, where $l_{a}^{2}=\dfrac{4bc\cdot s(s-a)}{(b+c)^{2}}\leq s(s-a)$
$\therefore h_{b}^{2}+h_{c}^{2}\leq l_{b}^{2}+l_{c}^{2}\leq s(s-b)+s(s-c)=sa$, where $h_{b}$ is $B$-altitude

$\Longrightarrow$ $h_{b}^{2}+h_{c}^{2}=4(\triangle ABC)^{2}\cdot\left(\dfrac{1}{b^{2}}+\dfrac{1}{c^{2}}\right)=\dfrac{abc}{R}\cdot rs\cdot \left(\dfrac{1}{b^{2}}+\dfrac{1}{c^{2}}\right)\leq sa$
$\therefore$ $\dfrac{R}{r}\geq \dfrac{b}{c}+\dfrac{c}{b}$

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  • $\begingroup$ Wow! How did you figure this out? $\endgroup$ – Sawarnik May 6 '14 at 9:54
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en, Notice that $$S=\frac{abc}{4R}, S=pr$$ where $p=\frac{a+b+c}2$.Heron's formula give $S=\sqrt{p(p-a)(p-b)(p-c)}$, we get that $$\frac Rr=\frac{2abc}{(a+b-c)(b+c-a)(c+a-b)}$$ then, take $a=y+z,b=z+x,c=x+y$, yields $$\frac Rr=\frac{(x+y)(y+z)(z+x)}{4xyz}$$ then, $$\frac bc+\frac cb=\frac{z+x}{x+y}+\frac{x+y}{z+x}$$ so, it is sufficient to show that $$(x+y)^2(y+z)(z+x)^2\geq 4xyz[(z+x)^2+(x+y)^2]$$ ha, $(x+y)^2z\geq4xyz$ give $$z(x+y)^2(z+x)^2\geq 4xyz(z+x)^2,$$ and $y(z+x)^2\geq4xyz$ yields $$y(x+y)^2(z+x)^2\geq 4xyz(x+y)^2$$ we get the result

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  • $\begingroup$ Thank you ,It's very nice.+1 $\endgroup$ – math110 Aug 11 '13 at 12:55

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