45413 三角形旁徑與高之間的三個性質

\begin{align*} (1)\ &\sum h_ah_b\le \sum r_ar_b&p_{275}\ \hbox{第 15 個不等式。} \\[8pt] (2)\ &\sum \dfrac{h_a+h_b}{r_a+r_b}\le 3&p_{276}\ \hbox{第 41 個不等式。} \\[8pt] (3)\ &\prod \dfrac{h_b+h_c}{r_a+h_a}\le 1&p_{276}\ \hbox{第 50 個不等式。} \end{align*}

(2) $\displaystyle\sum \dfrac{h_a+h_b}{r_a+r_b}=2+\dfrac{2r}R$.
(3) $\displaystyle\prod \dfrac{h_b+h_c}{r_a+h_a}=\dfrac{2r}R$.

$\therefore$ $r_a=\dfrac{rp}{p-a}$ $h_a=\dfrac{2rp}{a}$.

$\because$ $\displaystyle\sum a=2p$, $\displaystyle\prod (p-a)=r^2p$, $\displaystyle\sum ab=p^2+4R r+r^2$, $\displaystyle\prod a=4Rrp$.

(1) $\because$ $h_ah_b=\dfrac{4r^2p^2}{ab}$ $$r_ar_b=\frac{r^2p^2}{(p-a)(p-b)}=\frac{r^2p^2(p-c)}{\prod (p-a)}=\frac{r^2p^2(p-c)}{r^2p}=p(p-c)$$ $$\therefore\ \sum h_ah_b=\sum \frac{4r^2p^2}{ab}=4r^2p^2\sum\frac 1{ab}=4r^2p^2\cdot \frac{\sum a}{\prod a}=4r^2p^2\cdot \frac{2p}{4Rrp}=\frac{2rp^2}{R}$$ $$\sum r_ar_b=\sum\frac{r^2p^2}{(p\!-\!a)(p\!-\!b)}=\sum\frac{r^2p^2(p\!-\!c)}{\prod (p\!-\!a)}=\sum\frac{r^2p^2(p\!-\!c)}{r^2p}=p\sum(p-c)=p^2$$ $$\therefore\ \sum h_ah_b=\frac{2r}{R}\sum r_ar_b.$$

(2)

\begin{align*} \because\ \frac{h_a+h_b}{r_a+r_b}=&\dfrac{\dfrac{2rp}{a}+\dfrac{2rp}{b}}{\dfrac{rp}{p\!-\!a}+\dfrac{rp}{p\!-\!b}}=\frac{2(a+b)(p-a)(p-b)}{ab(2p-a-b)} =\frac{2(2p-c)(p-a)(p-b)}{\prod a}\\ =&\frac{2[p+(p-c)](p-a)(p-b)}{4Rrp}=\frac{(p-a)(p-b)}{2Rr}+\frac{\prod (p-a)}{2Rrp}\\ =&\frac{(p-a)(p-b)}{2Rr}+\frac{r^2p}{2Rrp}=\frac{(p-a)(p-b)}{2Rr}+\frac{r}{2R}\\ \therefore\ \sum\frac{h_a+h_b}{r_a+r_b}=&\sum\Big[\frac{r}{2R}+\frac{(p-a)(p-b)}{2Rr}\Big]=\frac{3r}{2R}+\frac{\sum p^2-p\sum(a+b)+\sum ab}{2Rr}\\ =&\frac{3r}{2R}+\frac{3p^2-4p^2+p^2+4Rr+r^2}{2Rr}=2+\frac{2r}{R}. \end{align*}

(3)

\begin{align*} \because\ \frac{h_b+h_c}{r_a+h_a}=&\dfrac{\dfrac{2rp}{b}+\dfrac{2rp}{c}}{\dfrac{rp}{p\!-\!a}+\dfrac{2rp}{a}}=\frac{2(b+c)a(p-a)}{(2p-a)bc} =\frac{2(b+c)a(p-a)}{(b+c)bc}=\frac{2a(p-a)}{bc}\\ \therefore\ \prod\frac{h_b+h_c}{r_a+h_a}=&\prod\frac{2a(p-a)}{bc}=\frac{8\prod a\prod(p-a)}{\prod bc}=\frac{8\cdot 4Rrp\cdot r^2 p}{(4Rrp)^2}=\frac{2r}{R}. \end{align*}

參考文獻

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