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
�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�5�
*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$íÞ�t�,
¥_½æœ�\7,�, .¬´u��ª©
í� Bb¡5-ÞíÇ 16£Ç17
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Ç 16
Ç 17
ûi$�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$ír�½æ, 7uÞ�½æ (‡6Ã
«, (6œ �: sûi$r�†Þ�ó�,
¥5.Í), FJ*s‘úi(~pœ�Ý�
Bb}Aú_¥�V25� ln�oûi$
í8$�
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Ç18
Ç19
(I) ˛øûi$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)„�
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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$íÞ�Ñ|×�
ü . !x
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ø…dín�R�ƒüi$í8$ÿ˛
%'˚Ø7.~õÒ (9õ,u•.¦)� Ç
øjÞ, R�ƒú&˛È, ¥}−£ƒÞ�D
ñ�íl�, H«ìÜD Ptolemy ìÜ6}
�ªø¥íR�, ¥