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1、數uni5B78傳uni64AD 35uni53774uni671F, pp. 82-85有uni95DCuni4E09角形內切uni5713等分線uni7684uni4E00些不等式莊健uni7965uni6458要: uni672C文uni5229uni7528uni4E09角形uni7684內切uni5713等分線uni9577uni70BAuni904Euni6E21uni91CF, 簡uni5316uni56DB個uni4E09角幾何不等式uni7684uni8B49明。uni95DCuni9375uni8A5E: uni4E09角形內切uni5713等分線, uni4E09角幾何不
2、等式。uni672C文uni7B26號uni5B9A義: a,b,c 表 ABC uni7684uni4E09邊uni9577; 表 ABC uni7684uni9762積; s 表 ABC uni7684uni534Auni5468uni9577; r 表 ABC uni7684內切uni5713uni534A徑; ma, mb, mc 表uni4E09中線uni9577; ta, tb, tc 表uni4E09內角平分線uni9577; ha, hb, hc 表uni4E09邊上uni7684uni9AD8; ra, rb, rc 表uni4E09傍切uni5713uni534A徑; la
3、, lb, lc 表uni4E09內切uni5713等分線uni9577。uni58F9、uni4E09角形內切uni5713等分線uni7684uni7531來1988 年第 29 屆uni570Buni969B奧uni6797uni5339亞數uni5B78uni7AF6賽 IMO uni958B賽uni524D, 依慣例uni5404uni570Buni9808提供uni9810uni9078uni984C給uni547Duni984Cuni59D4uni54E1會uni9078uni7528,其中第84 uni984C是uni7531uni7576時蘇聯所提供,其uni984Cuni7
4、6EEuni70BA:uni5728ABC uni7684BC 邊上uni53D6uni4E00uni9EDED,使得ABD uni548CACD uni7684內切uni5713uni534A徑uni76F8等,uni6C42uni8B49: AD2 = cot A2。(uni56E0 AD 使得ABD uni548CACD uni7684內切uni5713uni534A徑uni76F8等,而中文uni537Buni7121統uni4E00uni7684uni6B63式uni8B6Funi540D,uni56E0uni6B64uni7B46者uni6BD4uni7167uni4E09角形中邊
5、或角等分uni7684uni6982念, uni5C07uni5B83稱uni70BAABC uni5728 BC 邊上內切uni5713等分線。)uni8B49明 : 依 Heron 公式uni53CAuni4E09角形uni534A角uni5B9Auni7406uni77E5:cot A2 =radicalbigs(sa)(sb)(sc)radicalBiggs(sa)(sb)(sc) = s(sa),故uni53EA要uni8B49得:AD =radicalbigs(sa), uni6B64uni984Cuni5373uni53EF得uni8B49, 今uni5982下uni5716所u
6、ni793A:82有uni95DCuni4E09角形內切uni5713等分線uni7684uni4E00些不等式 83設ABD uni7684uni9762積uni70BA1, uni534Auni5468uni9577uni70BA s1, 內切uni5713uni7684uni5713心uni70BA O1,設ACD uni7684uni9762積uni70BA2, uni534Auni5468uni9577uni70BA s2, 內切uni5713uni7684uni5713心uni70BA O2,且其uni76F8等之內切uni5713uni534A徑uni70BAr; ABC uni
7、7684內心uni70BA I, 而 AD 設uni70BA la, uni5247有r = 1s1= 2s2= 1 +2s1 +s2= s+la rr =r(s+la) =ss+la; (1)uni53C8設ABD uni7684內切uni5713與 AB 切於 E, ACD uni7684內切uni5713與 AC 切於 M, 而ABCuni7684內切uni5713與 AB uni53CA AC 分uni5225切於 F uni53CA N, uni56E0 BO1E uni548C BIF uni76F8似, 且 CO2M uni548CCIN uni76F8似, 所以有:rr =BE
8、BF =s1 lasb =CMCN =s2 lasc =s1 +s2 2la2sbc =slaa ; (2)uni7D9Cuni5408 (1)、(2) 得:ss +la =slaa s2 l2a = as l2a = s2 as la =radicalbigs(sa)。上述之 AD = la =radicalbigs(sa) = 12radicalbig(a+b +c)(b+ca), 得uni8B49。uni8A3B:1. 有uni95DCuni4E09角形內切uni5713等分線uni76F8uni95DC性uni8CEAuni7684介uni7D39uni53CAuni53E6uni59
9、16uni5229uni7528 STEWART uni5B9Auni7406所得uni7684uni8B49明, uni8B80者uni53EFuni53C3uni770B 2。2. 2010 年 AIME 邀請賽第uni4E00uni968Euni6BB5uni7684第 5 uni984C也是uni95DC於uni4E09角形內切uni5713等分線uni7684uni984Cuni76EE, 22 年後uni53C8uni91CD新出uni73FEuni5728uni570Buni969B數uni5B78uni7AF6賽上, 令人驚uni559C。84 數uni5B78傳uni64AD
10、 35uni53774uni671F uni6C11100年12月uni8CB3、uni5229uni7528uni4E09角形內切uni5713等分線uni9577簡uni5316uni4E09角幾何不等式uni7B46者uni767Cuni73FEuni5C07內切uni5713等分線uni9577 la uni7576成不等式中uni7684uni904Euni6E21uni91CF, uni53EF簡uni5316uni4E00些uni904Euni53BBuni9808uni7E41uni8907技uni5DE7才能uni8B49明uni7684uni4E09角幾何不等式, 接下來u
11、ni8B49明uni4E00個uni91CD要uni7684uni9810備uni5B9Auni7406:uni5F15uni7406: uni5C0D任意ABC, ma la ta ha。uni8B49明 : uni56E0uni70BAma = 122b2 + 2c2 a2 = 12b2 + 2bc+c2 a2 +b2 +c2 2bc= 12radicalbig(b +c)2 a2 + (bc)2 = 12radicalbig(b+c+a)(b +ca) + (bc)2 la (3)uni53C8 ta = 2bcb+c cos A2 = 2bcb +cradicalbigs(sa) =
12、2bcb +cla la。 (4)uni7531 (3)、 (4) uni53CA ta ha uni53EFuni77E5:uni5C0D任意ABC, ma la ta ha, 而等號成立於 b = c 時,亦uni5373ABC uni70BAuni4E00等腰uni4E09角形時: ma = la = ta = ha。以上述uni5F15uni7406uni70BA基uni790E, 並以內切uni5713等分線uni9577 la uni70BAuni904Euni6E21uni91CF, 我們uni53EF得出下列uni56DB個不等式, 分述於uni5982下uni7684uni5
13、6DB個uni5B9Auni7406:uni5B9Auni7406uni4E00:summationdisplaym2a s2 summationdisplayt2a summationdisplayh2a, 其中summationdisplaym2a 代表 m2a, m2b, m2c uni7684迴圈uni548C, 亦uni5373summationdisplaym2a = m2a +m2b +m2c, 其uni9918uni985E推。uni8B49明 : uni56E0uni70BAsummationdisplayl2a =summationdisplays(sa) = s2, 且u
14、ni7531uni5F15uni7406,uni5373得:summationdisplaym2a summationdisplayl2a = s2 summationdisplayt2a summationdisplayh2a。uni5B9Auni7406uni4E8C: mambmc rarbrc tatbtc hahbhc。uni8B49明 : uni56E0uni70BArarbrc =parenleftBig saparenrightBigparenleftBig sbparenrightBigparenleftBig scparenrightBig= 3(sa)(sb)(sc) =
15、 rs2uni53C8lalblc =radicalbigs(sa)s(sb)s(sc) = s = rs2 = rarbrc,且uni7531uni5F15uni7406,uni5373得: mambmc lalblc = rarbrc tatbtc hahbhc。uni5B9Auni7406uni4E09:summationdisplaymata summationdisplayrarb。有uni95DCuni4E09角形內切uni5713等分線uni7684uni4E00些不等式 85uni8B49明 : uni9996先uni8B49明mata l2a parenleftBig122b
16、2 + 2c2 a2parenrightBigparenleftBig 2bcb+c cosA2parenrightBig s(sa)parenleftBig2b2 + 2c2 a2parenrightBigparenleftBig 1b +c cosA2parenrightBig s(sa)bc = cos2 A2 ,2b2 + 2c2 a2b +c cosA2 , b2 +c2 + 2bccosA(b+c)2 cos2 A2 =1 + cosA2 ,2b2 + 2c2 + 4bccosA (b +c)2 + (b +c)2 cosA,2b2 + 2c2 (b+c)2 (b+c)2 4bccosA,(bc)2 (bc)2 cosA。等號成立於 b = c 時, (5)un
