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四边形的面积

2009-03-12 12页 pdf 430KB 38阅读

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四边形的面积 ûi$íÞ� ��p ø . ½æíT| #ø_úi$, ˛øúiÅ, µó…í Þ�ªàO±í Heron t�V°��¥Bb Ê“} Heron t�: pø¨`ç%ð”ød 2, Ð%�FÅH[1]� ÛÊb‹JR�, Bb AÍ;ƒ7s_j²: (i) &bíTò, *� Þ½æ‰A˛È½æ; (ii) ibíÓ‹, *ú i$‰Aûi$4ByÖi$� …dcÌk n�R�ƒûi$í8$� ½æ: ˛øûi$íûi a, b, c, d, � ÌéNk Heron íÞ�t�? Êd.,, ...
四边形的面积
ûi$íÞ� ��p ø . ½æíT| #ø_úi$, ˛øúiÅ, µó…í Þ�ªàO±í Heron t�V°��¥Bb Ê“} Heron t�: pø¨`ç%ð”ød 2, Ð%�FÅH[1]� ÛÊb‹JR�, Bb AÍ;ƒ7s_j²: (i) &bíTò, *� Þ½æ‰A˛È½æ; (ii) ibíÓ‹, *ú i$‰Aûi$4ByÖi$� …dcÌk n�R�ƒûi$í8$� ½æ: ˛øûi$íûi a, b, c, d, � ÌéNk Heron íÞ�t�? Êd.,, ¥˛%� Brahmagupta t � (628�, ��bçð) £ Bretschneider t� (1842 �)� .¬…dÉ~í�-½æ u: J©� 25í¬˙, ?¹àS“¿|t �? |ßu?“¯ÜË”õ|V� Bbı�® ƒ¥šíñ™:#B“ 4íim” (insight), .bÉ#B“j4Dbå”� ¥uÞúbçv, ø_!…7à‰íè�� .�uúi$Cûi$,Éki�i� ú i(£Þ�5ÈíÉ[, �s_½bí!‹: ø_ui� i� úi(íÉ[�, WàH« ìÜ (¼òìÜ)� ìýìÜD Ptolemy ì Ü; Çø_uÞ�[Ai� iCúi(5t �, Wà Heron t�, Brahmagupta t �D Bretschneider t�� ¥<ìÜDt� íÉ[Ý ò~, x�øñsÞíX-É©, �òu˘k°øðí, ĤBbbø9‹J n�� …bªJzu©túiç��Þ Sç 21(í45,IA¼©2â.�„G� 9õ ,, Sçí²¾Hb“ÿuJ¥<Ö‡T Ñ25í�œD|êõ� úkò2ÞVz,¥ uø_›*25íß�æ� ù . úi$íÅ]øh |O±7�øíuÉkòiúi$í! ‹: H«ìÜ:� C = 90◦ ⇐⇒ c2 = a2+b2 Þ�t�:S = 1 2 ab, ¡c-Ç1� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 1 Ç 2 Å: H«ìÜÊd.,� 370 �„¶ [5]� 1 2 bçfÈ �þ»ú‚ ¬82�9~ QOì&ƒøOúi$, ¤vªœîó Ö˘� H«ìÜR�A ìýìÜ:   a2 = b2 + c2 − 2bc cosA b2 = c2 + a2 − 2ca cosB (i� iÉ[) c2 = a2 + b2 − 2ab cosC (1) Þ�t�: S= 1 2 ab sinC = 1 2 bc sinA = 1 2 ca sinB (2) C S = √ s(s− a)(s− b)(s− c) (Heron t�) (3) w2 s = 1 2 (a + b + c)� àSòhË“| Heron t�ín�~ ¡c [1]� ,H (3) �í„p, É.¬u (1)�(2) s�í�À@à: S = 1 2 ca sinB (â(2) �) 16S2 = 4a2c2 sin2 B = 4a2c2(1− cos2 B) = 4a2c2[1− (c 2 + a2 − b2 2ac )2] (â(1) �) = 4a2c2 − (c2 + a2 − b2)2 I s = 1 2 (a + b + c), † S2 = s(s− a)(s− b)(s− c), „H� ú . R�ƒÆqQûi$ ûi$¥øŪúi$Å´bÑÜ� � á (delicate, subtle)� |éOíuûi$³ � �4: ˛øûi a, b, c, d, 1³�ñø ²ìø_ûi$ (Sàr�hõ), …´uª J9ò� Tií0²7‰$� WàÅj$ªJ T®�‰$ (ûi\M.‰): � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 3 ¥û_ûi$·.r�, 1/Þ�·. °� ½1. ˛øûi$ûiÑ a, b, c, d, (i) i� i� úi(�SÉ[? (ii) Þ�àS[Ai� iCúi(? Obç25í Ü,Bbl¢ƒÔW,y M¥©ä«�,‡ªƒøO8$� Bóuûi $í�ÀÔWá? Bb'AÍË;ƒ7Åj $: ûi$íÞ� 3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 4 …âs_ó°íòiúi$ a, b, c EÊø –, ĤiDúi(íÉ[EÍÉuH«ì Ü: c2 = a2 + b2, 1/Þ� S = ab� ¥ <·³�h2� àS““4ÖÑÿJ”á? à‹Bbø,HÅj$íiT0²A+ $: � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 5 µóÞ� S = 1 2 xy; OuiDúi(íÉ [EÍ.qõ|V�9õ,, +$ªJÊûi \M.‰5-, T9òC…., éúi( x, y ‰�� Ĥ´u�õËÍ.qzší>g� Bbø−LSúi$.ªqQkø_Æ52, ûi$†.Í� Bbš¢ø¥: 5?ÆqQä$D+$ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç6 � � � � Ç 7 Ç7 +$íÞ�Ñ S = 1 2 xy 7iDúi(íÉ[uBóá? éÍÇ6íä$DÇ7 í+$x�ó° íÞ� (JÆ-©QûÝõ�ø), ] 1 2 xy = ab ÄÑ a = c, b = c, FJ xy = 2ab = ab + cd (4) ?¹súi(í���ksúi��5¸� ¥ÿuÆqQ+$íiDúi(íÉ[�� J¥_t�VhõÇ6, BbêÛ αβ = ac + bd (5) 6A�, ÄÑ (5) �.¬uH«ìÜ α2 = a2 + b2 5“sj“”C “si“”(ÄÑ α = β, a = c, b = d)� Ĥ (4) �ªJõTuH«ìÜ íø�R�� 4 bçfÈ �þ»ú‚ ¬82�9~ ,H!‹óýBb“¿: ÆíL AB k E õ, U) � CDE = � BDA� kuñqõ| �CDE ∼ �BDA1/�ADE ∼ �BDC *7 CD BD = CE AB , BC AE = BD AD ku CD · AB = BD · CE AD · BC = AE · BD s�ó‹) CD · AB + AD · BC = BD(AE + CE) = BD · AC ?¹ ac + bd = xy. (6) „H� Ĥ, Bb)ƒ: ìÜ1: (Ptolemy, 150 �) q ABCD ÑÆqQûi$, ûi}�Ñ a, b, c, d, ú i(Ñ x, y, † xy = ac + bd� ¥_!‹�Í1\, ¢uH«ìÜíR �� Ùdçð Ptolemy (90-168�) ‚à… d|vÍ, ø"ýƒb[� FúÙdçÝ #Ï,Fz¬: “Cü�{íB,…@à.9 øO�Þ˛§� Ou©çBcƒÅÙõrÊ ˛2YÎAÐí*−�Í�åË«Wv, ÿ 8.AŠ�™ÊÙ,AÈ5>, ßduÙÿ ‹g (Zeus) fAÇBJÿ†�” öuIA> �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 9  � (1992�) ×ç:5AÍ �ø5 æà-: ÊÇ 9 2, AD ÑÆ5ò�, B, C њƶ,sõ� a = AB, b = BC, c = CD, d = AD, t„ d Ñj˙� x3 + (a2 + b2 + c2)x− 2abc = 0 5ø;� ¥çÍ���„¶, Ou‚à Ptolemy ìܺ¯H«ìÜíd¶u|�ÝÔ�í� I x = BD, y = AC, † xy = ac + bd ûi$íÞ� 5 x2 = d2 − a2 y2 = d2 − c2 ku (d2 − a2)(d2 − c2) = x2y2 = (ac + bd)2 �Ç� “�) d3 − (a2 + b2 + c2)d− 2abc = 0 ¥ÿ)„7� ;W°6©»í%ð, ³�õƒ5ÞS à,H„¶� úí5ÞÖšuSà: T^Œ (DìýìÜVd, œ�¦� QOBb«©ÆqQûi$íÞ�t �, EÍ¡5Ç 8� ílhôƒ, Þ�âûi a, b, c, d ñø²ì� ûi$íiT0²ª? �àr�, O1.�àÞ�� ĤÆqQûi $íÞ�Ü@�ú@í Heron t�, BbI wÞ�Ñ S(a, b, c, d)� ½ 2. S(a, b, c, d) =? Bbªø¥hôƒ S(a, b, c, d) x�- �4”: (i) S(a, b, c, d)í¾å (dimension) Ñ L2 (¹Å�í�j), (ii) iä‘K:ç a+b+c = dC b+c+d = a C c + d + a = b C d + a + b = c v, S(a, b, c, d) = 0, ]âÄ�ìÜ ø S(a, b, c, d) � (a + b + c − a), (b + c + d − a), (c + d + a − b) D (d + a + b − c) 5Ää� û6�–V, ¾åÑ L4� ;W¥s‘(Ø, óýBbT|-Þí “¿: S2 = K(a + b + c− d)(b + c + d− a)· (c + d + a− b)(d + a + b− c) w2 K u&ì b� J£j$5ÔWHp, �, �¹°) K = 1 16 � kuBbí“¿êr p�: S2 = (s− a)(s− b)(s− c)(s− d) (7) w2 s = 1 2 (a + b + c + d)� ¥uBbFbí �ý? BbtðÅj $, êÛ (7) �A�� úk d = 0 5ÔW, ûi$‰Aúi$,7 (7) �‰A Heron t �� Ĥ, Ê´³�„p5‡, Bb˛%�7 óçíÜâó] (7) �ÿuÆqQûi$í Þ�t�� ´„C„p, b•¨ø‘˜? éBbþ t„pß� EÍ¡cÇ 8� ûi$íÞ� S = �ABC +�ACD = 1 2 ab sinB + 1 2 cd sinD 4S = 2ab sinB + 2cd sinD (8) âìýì�ø a2 + b2 − 2ad cosB = y2 = c2 + d2 − 2cd cosD FJ a2 + b2 − c2 − d2 = 2ab cosB − 2cd cosD (9) 6 bçfÈ �þ»ú‚ ¬82�9~ ø (8), (9) s��jó‹) 16S2 + (a2 + b2 − c2 − d2)2 =4a2b2 + 4c2d2 − 8abcd cos(B + D) (10) ÄÑ B + D = 180◦, cos(B + D) = −1, ]) 16S2+(a2+b2−c2−d2)2 = (2ab+2cd)2 *7 16S2 = (2ab + 2cd)2 − (a2 + b2 − c2 − d2)2 = [(a + b)2 − (c− d)2]· [(c + d)2 − (a− b)2] = (a + b + c− d)(a + b− c + d) (c + d + a− b)(c + d− a + b) I s = 1 2 (a + b + c + d), †) S2 = (s− a)(s− b)(s− c)(s− d) Bbí“¿)„� ì Ü 2:(Brahmagupta,628 �) qABCD ÑÆqQûi$, ûiÑ a, b, c, d, †wÞ�Ñ S = √ (s− a)(s− b)(s− c)(s− d)� (10) Å. Brahmagupta ÏJѤt�_à kLSûi$� 9õ,, Heron ˛N|øO ûi$̶âwûiñø²ì� û . øOûi$ øOûi$ABCDª}Aoûi$ (convex quadrilateral), à-Ç 10, J£ pûi$, à-Ç 11� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 10 Ç 11 Bbln�oûi$í8$� àÇ 10, q ABCD Ñø_oûi$, 1/ AB = a,BC = b, CD = c,DA = d AC = x,BD = y, s = 1 2 (a + b + c + d), S = ABCD íÞ�� Bbbû˝í�æEÍu½ 1, íl« nûi a, b, c, d Dúi( x, y íÉ[� ú kÆqQûi$í8$, Ptolemy ìܵs ûi$íÞ� 7 Bb: xy = ac + bd� OuúkøOoûi $, àSá? øÆqQûi$šT9ò, ûi\M. ‰, ¹ ac + bd .‰, Ouúi( x D y º ø_‰Å, Çø_‰s , pÑ x0 D y0� µ ó x0y0 D xy S6œ×á? N˛.ñqõ |V, öÜR)ªœ¿7 (OuBbó]�ö ܪ©)� -ÞBbbSàF‚í“«Ø4í ÆU¶”, _"Ÿl Ptolemy ìÜí„pj ¶, ttõ})ƒBó!�� �ƒÇ10 5øOoûi$� T|õ E, U) � DAE = � CAB / � ADE = � ACB ku �ADE ∼ �ACB, ] AD ED = AC BC , ¹ bd = x · ED (11) ÇÕ6� AB AE = AC AD , 1/ � DAC = � EAB *7 �ABE ∼ �ACD, ] AB BE = AC CD , ¹ ac = x · BE (12) (11)+(12) ) ac+bd = x·BE+x·ED = x·(BE+ED) ÄÑ DE + ED ≥ BD = y, ] xy ≤ ac + bd. (13) Ê,HÆU¬˙2,Bb6êÛ: (13)�2í �UA�5kb‘Ku E rÊúi( BD ,, ¹ A, B, C, D ûõOåuÆ� úkpûi$í8$, (13) �6A�� à-Ç12, ø AB, BC ú AC TŸ¦, ) ƒoûi$ AB′CD, I B′D = y′, †â, H„pø xy′ ≤ ac + bd ÄÑ y ≤ y′, FJ xy ≤ ac + bd. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 12 ªø¥, Bbhô �ÔæD¢“í8 $ (Ì8~¡c [2]): (i) çûi$ ABCD p…AX $v (Ç 13), (13) �6A�, � � � � � � � Ç 13 8 bçfÈ �þ»ú‚ ¬82�9~ (ii) çûi$ ABCD ¢“Aúi$v, . �u�sõ½¯Cw2øõrÊø_i , (Ç14), (13) �EÍA�� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 14 (iii) çûi$ABCD ¢“AÊøò(, v, (13) �A�, 1/çûõÊò(, O A, B, C, D 5ßå§�v, (13) � ‰A��: ¹ AB ·CD+BC ·AD = AC · BD� (Euler ìÜ)� ,!,H5n�, Bb)ƒ ìÜ3: (R�í Ptolemy ìÜ, ÿ�) úk�Þ,L<ûõ A, B, C, D, - �0A�: AB · CD + BC · AD ≥ AC ·BD 1/�UA�íkb‘Ku A, B, C, D û õOåuÆCOåu(� Å.ò(DÆx�°�ËP,ò(ux� ̤ך�5Æ� *òiúi$íH«ìÜ: c2 = a2+b2, ƒL<úi$í c2 ≤ a2 + b2 (ç � C ≤ 90◦) C c2 ≥ a2 + b2 (ç � C ≥ 90◦), �� üíìýìÜ: c2 = a2+b2−2ab cosC� ° Ü, *ÆqQûi$í Ptolemy ìÜ: xy = ac+ bd, ƒL<ûi$í xy ≤ ac+ bd, @ v6�ó@í�ü��ß? �ƒÇ 10 5øO oûi$� â k�ADE ∼ �ACB / �ABE ∼ �ACD, ] � AED = � B / � AEB = � D � BED = 2π − ( � AEB + � AED) = 2π − ( � B + � D) âìýìÜø y2=ED2+BE2−2ED·BE cos( � BED) y2=ED2+BE2−2ED·BE cos(B + D) si°�J x2 ) x2y2 = (x · ED)2 + (x · BE)2 −2(x ·ED)(x ·BE) cos(B + D) yâ (11) £ (12) �) x2y2 = a2c2 + b2d2 − 2abcd cos(B + D) (14) ÄÑ A + B + C + D = 360◦, ]6� x2y2 = a2c2 + b2d2 − 2abcd cos(A + C) (15) Bb·<ƒs_ÔW: (i) ç B +D = 180◦ v, ?¹ A, B, C, D ûõuÆv, (14) ûi$íÞ� 9 C (15) �“�A Ptolemy ìÜ: xy = ac + bd� Ĥ (14) C (15) �̪eÑ Ptolemy ìÜíR�� (ii) ç B+D = 90◦ v, (14) C (15) �“�A x2y2 = a2c2 + b2d2 (16) ÇÕ, (14) �úkpûi$6A�, w „pÉb¡5-ÞÇ 15, 7¬˙êr�,H oûi$í�„øš� � � � � � � � � � Ç 15 ìÜ4: (R�í Ptolemy ìÜ, #�) úkL<íûi$, àÇ 10 CÇ 11, 0� x2y2 = a2c2 + b2d2 − 2abcd cos(B + D). úkL<íûi$, éÍâ (14)�ª) (13) �, ?¹âìÜ4ªR|ìÜ3� |(BbJ©L<ûi$íÞ�t�, ¥_½æœ�\7,�, .¬´u��ª© í� Bb¡5-ÞíÇ 16£Ç17 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç 16 Ç 17 ûi$�û_i a, b, c, d, û_i � A, � B, � C, � D, J£s‘úi( x, y, ,u� 10 _bÖ, …b1ÝêrÖ�, WàBb�#� íR�í Ptolemy ìÜJ£û_i5¸Ñ 360◦, ¥s6·uúk 10 _bÖíÌ„‘ K� ûi.—J²ìûi$í$Õ, ¥uc _½æí,�FÊ� ÄÑBbbJ©í.u ûi$ír�½æ, 7uÞ�½æ (‡6à «, (6œ �: sûi$r�†Þ�ó�, ¥5.Í), FJ*s‘úi(~pœ�Ý� Bb}Aú_¥�V25� ln�oûi$ í8$� 10 bçfÈ �þ»ú‚ ¬82�9~ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ç18 Ç19 (I) ˛øûi$ís‘úi( x, y, 1 /…b�ó�ò, ¡cÇ 18� éÍûi$í Þ�Ñ S = 1 2 xy (17) (II) ˛øúi( x, y £…bíHi θ, ¡cÇ 19� Iûi$íúi( x = x1 +x2, y = y1 + y2, kuûi$íÞ�Ñ S=�AOB+�BOC+�COD+�DOA = 1 2 x1y1 sin θ + 1 2 x1y2 sin(π − θ) + 1 2 x2y2 sin θ + 1 2 x2y1 sin(π − θ) S= 1 2 xy sin θ (18) ç θ = 90◦ v, (18) �“�A (17) �� (III) yªø¥, ¢ø−ûi, ?¹˛ø a, b, c, d, x, y� ¤vûi$ñø²ì7� à S°Þ�á? (18) �EÍA�, OuàSø sin θ ¾jA a, b, c, d á? ¥UBb;ƒ7 ìýìÜ� â (18) �) 16S2 = 4x2y2 sin2 θ = 4x2y2(1− cos2θ) = 4x2y2 − (2xy cos θ)2 ¢ÄÑ (¡cÇ 19) 2xy cos θ=2(x1 + x2)(y1 + y2) cos θ =2x1y1 cos θ + 2x1y2 cos θ +2x2y2 cos θ + 2x2y1 cos θ =2x1y1 cos θ − 2x1y2 cos(π − θ) +2x2y2 cos θ − 2x2y1 cos(π − θ) =(x21 + y 2 1 − a2)− (x21 + y22 − b2) +(x22 + y 2 2 − c2)− (x22 + y21 − d2) =−a2 + b2 − c2 + d2 FJ 16S2=4x2y2 − (a2 − b2 + c2 − d2)2 (19) Bb6ªø (19) �, N¬ºj, ZŸÑ 16S2=(2ac + 2bd)2 − (a2 − b2 + c2 − d2)2 −4(ac + bd)2 + 4x2y2 =(a + b + c− d)(b + c + d− a) (c + d + a− b)(d + a + b− c) −4[(ac + bd)2 − x2y2] S2=(s− a)(s− b)(s− c)(s− d) −1 4 [(ac + bd)2 − x2y2] (20) ûi$íÞ� 11 â¤�¹õ|: A,B,C,DûõOåuÆ ⇐⇒xy = ac + bd (Ptolemy ìÜ) ⇐⇒S2 = (s− a)(s− b)(s− c)(s− d) (Brahmagupta t�) M)Ô�·<íu, Ê,Húkoûi $íú¥��„2, (17)�(18)�(19)�(20) û _t�úkpûi$EÍA�� ¥Éb¡5 -Ç 20 1/",H�„¹ª)„� � � � � � � � � � � � Ç 20 QO, N¬#�íR�í Ptolemy ì Ü, (19) �ªJªø¥ZŸAà-: 16S2 = 4(a2c2 + b2d2 − 2abcd cos(B + D)) −(a2 − b2 + c2 − d2)2 = (2ac + 2bd)2 − (a2 − b2 + c2 − d2)2 −8abcd(cos(B + D) + 1) = (a + b + c− d)(b + c + d− a) (c + d + a− b)(d + a + b− c) −16abcd cos2(B + D 2 ) S2 = (s− a)(s− b)(s− c)(s− d) −abcd cos2(B + D 2 ) (21) éÍ6� S2 = (s− a)(s− b)(s− c)(s− d) −abcd cos2(A + C 2 ) (22) ,!,Hn�, Bb)ƒ ìÜ 5:(Bretschneider t�,1842�) úkL<ûi$ (.�po), wÞ �SÑ 16S2 = 4x2y2 − (a2 − b2 + c2 − d2)2 C S2 = (s− a)(s− b)(s− c)(s− d) −1 4 [(ac + bd)2 − x2y2] C S2 = (s− a)(s− b)(s− c)(s− d) −abcd cos2(B + D 2 ) R� 1. çûi$ÑÆÕ~ûi$v,† S = √ abcd sin( B + D 2 ). R� 2. çûi$muÆqQ6uÆÕ ~ûi$v, † S = √ abcd R� 3. Êûi a, b, c, d #ìí8$ -, JÆqQûi$íÞ�Ñ|×� ü . !x 12 bçfÈ �þ»ú‚ ¬82�9~ ø…dín�R�ƒüi$í8$ÿ˛ %'˚Ø7.~õÒ (9õ,u•.¦)� Ç øjÞ, R�ƒú&˛È, ¥}−£ƒÞ�D ñ�íl�, H«ìÜD Ptolemy ìÜ6} �ªø¥íR�, ¥
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