View of Od hipotezy do twierdzenia

(1)

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^`_bac dbe&f:g\hjiDkl:mn_poqkr7sPtvu_biDu_wlbxByp_bz{|fbz/apkoqiDk ~}W$EB!M}$DW$w}$p$$

∗

jqE+5Dq!& ¢¡b£j+5¤Dq¥¦¥B£q§Z7¨7©ª7w` «+¬j7£j©ª& ¦¢q¨7£jw7w¢¡E5 q®¯5£q ¥%¯ 7¥7w &°5q©ª5¨7¥%£q§¥®D±¯7w¥7²¯ ¢q¥£j5¦5¬¯¡D³%5¦A!& ©ª¦¥7¡5¦A5"´5 ¥ 5¨q7j¡!£$§µ£q ©®5¨`$w«¯!£q57¥¦¥7¡bqD±¶§µ®¯ 5& 1£$´DD±$¯ £w£q§·p¸P57¥¥7w¦q¨ §¹q&5& $¦¥BD`B5ª¥q©ª$w«¯!£q57¥¦¥B¢q¨¢j±6¥®D±¯7w¥"£A§µ£j¨¨£¢³º±¯¦»E& 7w !£j¥¥¦¤5¨L³Z7«¯¥%£q§ ¢q¥£j5¦5¬¯·

ç

∗

!"#!%$'& & (*),+-.(/1032 & 4( & 57689 & :;6<=>*? ! @ /BACD & 5DE & #6FG & 5DIH4( & 57J,K

(2)

û

n

ÕÇwÔ¯Â`ÄÁLÝ¿¯Ñ&ÁqÊAÃD¾DÊ×p¿¯ÇDÑ&¾DÌ"ÅªÍÄåB½qÁ ÐÑ&¿!Ò!ÇDÄÁqÊ ¿¯ÇDÑ&åqÞ¯Ó+ÌL×EÄµÍ&Å|ÃEÁqÞ¯¿)ÌÎË&ÁqÃ

n

ÕÇwÔwËjæë\ÇwÅ|½qÓwÈÁªÍÄå¯Ø!âqÁª×EÑ7ÅÖÌBÒ!½qÄÌ"ÅLÈÁjÍËBÉEÄ×p¿|Ë&Áq½jÅEØ âqÁÈÁjÍËZË&¿ªÂ`ÁjÂ7ÉbÅÂ7ÉbÅ|Ñ7Å|Ç5Ë&ÁqÑ&¾!ÍË¾!Â`½qÃbÅ

n

ÕÇwÔwË7ÅBÝ¿¯Ñ&ÁqÊAÃEÁqÞ¯¿bØÖÓE½q¾!ÍÇDÓwÈÁqÊ-¾$ÌLÄåjÂ1ÃbÅ¯ÍË&åq×EÓwÈÔ¯Â`Á ËÌLÄÁqÑ7Ò!½qÁqÃEÄÁ;q t,uvuwxCyz|{};u~_{~z|{|3xN,3u7{3

n

~;;xu19{~zkxuyz|{}~xu{~z| d3xN,3u7{/u

n

~;/{1xuw{~;3t,u3{z|u_9; þ%ÿÿ

ç

, #¡v_vu¢z|¡/t|~;;xu'3z|//t,;u,s{};x£¤~xE}1^xu9;¥¦}¡¥pt,u7{~;/¡

û

A

c

B

b

a

C

D

E

α

β

§¨^©|ª¤«7¬®¯

(3)

°

û Ñ&½q¾¶¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅ¯Â7ÉÈÅ|ÇAÃbÅ-Ñ&¾!ÍÓEÃEÇDÓï½jÅDÏ¹à¯âqÊ-¾)ÌLÄåjÂ|ØDâqÁ

|AD| = |BE|.

û Ñ&½q¾jÈÊAÄÈÕ Ê-¾Ë&Áqâ¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅq

|AB| = c, |AC| = b, |BC| = a, p =

a+b+c

2

.

¶ ¿¯âqÆÄÌ1Á-Í&ÔAË&ÁqÑ7Å|½)Ò!ÌLÄÁ)Ò!Ñ&¿¯Þ¯ÄqbÍË7ÅÖÌLÄµÅ|Ê-¾/ÉEÄ×p¿|Ë&Áq½qå-Ä×EÑ&à¯ÀEÓwÈÁqÊ-¾-ÈÔ½qÌ1ÁqÑ&¾5ÙbÇ¯¿wÌ"Å¯ã ÆÓEÀ+é¹ÌL¾DÊ-ÓbÍ½jÅ|Ê-¾¯ê\Ë&Áq½qåª½×EÑ&½q¾jÈå`Ë¾!Â7É/½jÅDÏ¹¿¯âqÁqÚ:æ ·¸¹dº;»¼½¹^¾-ÿ ÄÆÓbÍË&Ñ&ÓwÈÁ$Ò!Ñ&ÓEÞ5Ô-½Ë¾!Â7É6Ò!Ñ&à¯Þ9q ø Á½qÃbÅ|ÃD¾!Â7É/ÌL½q¿¯Ñ&àwÌ'q

|AD

0

_{| =}

2 b+c

pbcp · (p − a), |BE| =

2 a+c

pacp · (p − b).

¶ Å|Ê-¾Ç¯¿¯ÆÁÈÃE¿9q

2 b

+ c

p

bcp

· (p − a) =

2 a

+ c

p

acp

· (p − b),

(a + c)

2

bcp

· (p − a) = (b + c)

2

acp

· (p − b),

(a

2

+ 2ac + c

2

)(bp − ab) = (b

2

+ 2bc + c

2

)(ap − ab),

a

2

bp−a

3

b

+ 2abcp − 2a

2

bc

+ c

2

bp

− c

2

ab

= b

2

ap

− ab

3

+ 2abcp − 2ab

2

c

+ c

2

ap−c

2

ab,

abp

· (a − b) − ab · (a

2

− b

2

) − 2abc · (a − b) + c

2

p

· (b − a) = 0,

(a − b) · (abp − ab · (a + b) − 2abc − c

2

p) = 0,

(a − b)

ab · (p − (a + b) − 2c) − c

2

_{p = 0,}

(a − b)

ab · (p − (a + b + c) − c) − c

2

p = 0,

(a − b)(ab · (−p − c) − c

2

p) = 0,

a

− b = 0,

a

= b.

¿ ½q¾!ÍÇwÅ|ÆÄµÐÊ-¾ÌLÄåjÂ ¹^¾ÀG· Ë&Áq½q¾3q z|¡/t|~;dt,uz|¡13{z|_xu99; ÁÂ¹^¾ÀG··Ã û Ñ&½q¾jÈÊAÄÈÊ-¾+¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅBÈÅ|ÇÃbÅAÑ&¾!ÍÓEÃEÇDÓïªÄv×p¿¯ÃbÅ¯ÒDË&¿

|AD| = d, |BE| = d.

ç ¾DÇwÅ|âqÁqÊ-¾¯Ø!âqÁ

2α = 2β.

ç ÁqäqÊ-¾×p¿!Ò/ÓDÌ"Å|Þ¯å\Ë&Ñ&à|ÈÇwÔwË

ABD;

½ËÌLÄÁqÑ7Ò!½qÁqÃEÄµÅAÍÄÃDÓbÍàwÌÎÒ!¿5ÍË7ÅwÈÁqÊ-¾3q

d

sin 2β

=

c

sin |^ADB|

,

Å)×p¿¯ÃEÄÁqÌ"Å|â

|^ADB| = π − 2β − α,

Ë&¿

d

sin 2β

=

c

sin (2β + α)

.

û ¿!Ò!¿¯ÀEÃEÄÁªÀEÄ¿¯Ñ7Å¯Â\×p¿!Ò/ÓDÌ"Å|Þ¯å\Ë&Ñ&à|ÈÇwÔwË

ABE

Ò!¿5ÍË7ÅwÈÁqÊ-¾

d

sin 2α

=

c

sin(2α + β)

.

(4)

Ü5Ë7Ô¯Ò

sin 2α

sin(2α + β)

=

sin 2β

sin(2β + α)

Ä:Ç¯¿¯ÆÁÈÃE¿

sin 2α sin(2β + α) = sin 2β sin(2α + β),

−2 sin 2α sin(2β + α) = −2 sin 2β sin(2α + β),

cos(3α + 2β) − cos(α − 2β) = cos(3β + 2α) − cos(β − 2α),

cos(3α + 2β) − cos(3β + 2α) = cos(α − 2β) − cos(β − 2α),

−2 sin

5α + 5β

2 sin

α

− β

2 = −2 sin

−α − β

2 sin

3α − 3β

2 .

Ü5Ë7Ô¯Ò4Ä1½qÁÌL½q¿¯Ñ&àwÌ'q

sin 3x = 3 sin x − 4 sin

3

x,

sin 5x = sin

5

x

− 10 sin

3

x

cos

2

x

+

5 sin x cos

4

x

Ò!¿5ÍË7ÅwÈÁqÊ-¾

sin

α

− β

2 sin

α

+ β

2 sin

4α + β

2 − 10 sin

2

α

+ β

2 cos

2

α

+ β

2 + 5 cos

4

α

+ β

2 =

= − sin

α

+ β

2 sin

α

− β

2 3 − 4 sin

2

α

− β

2 .

û Ñ&½q¾D×EÓbÐ&ã`Ê-¾¯Ø!âqÁ

α

6= β.

ç Ë&ÁjÒ!¾

sin

α

+ β

2 sin

α

− β

2 6= 0

Ä:Ç¯¿¯ÆÁÈÃE¿

sin

4

α

+ β

2 − 10 sin

2

α

+ β

2 cos

2

α

+ β

2 + 5 cos

4

α

+ β

2 = 4 sin

2

α

− β

2 − 3,

5 cos

2

α

+ β

2 − sin

2

α

+ β

2

− 4 sin

4

α

+ β

2 = 4 sin

2

α

− β

2 − 3,

5 cos

2

(α + β) + 3 = 4

sin

4

α

+ β

2 + sin

2

α

− β

2 .

û ¿¯ÃEÄÁqÌ"Å|â

0 < 2α + 2β < π,

½jÅwË&ÁqÊ

0 < α + β <

π

2

,

0 <

α+β

2

<

π

4

,

|

α−β

2

| <

π

4

¿¯Ñ7Å|½

5 cos

2

_{(α + β) + 3 > 3}

Ä

4 sin

4 α+β

2

+ sin

2 _α−β

2

<

4

√

2

4

+

√

2

= 3.

ëBË&Ñ&½q¾DÊ-ÓwÈÁqÊ-¾+Í×EÑ&½qÁjÂ`½qÃE¿5Ð&ã|æ ø ÅwË&ÁqÊ

α

= β, 2α = 2β, a = b.

ÁÂ¹^¾ÀG···Ã

A

c

B

x

z

y

C

D

E

α

β

§¨^©|ª¤«7¬_Å7¯

(5)

° û Ñ&½q¾jÈÊAÄÈÊ-¾?¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅAÈÅ|Ç?ÃbÅÑ&¾!ÍÓEÃEÇDÓ~õ!Ø

|AE| = y, |ED| = z

Ø

|BD| = x.

í\¿¯Ñ&½q¾!ÍË7ÅwÈÔ¯ÂB½ËÌLÄÁqÑ7Ò!½qÁqÃEÄµÅAÂ`¿5ÍÄÃDÓbÍàwÌÎÒ!¿5ÍË7ÅwÈÁqÊ-¾3q

c

2

= d

2

+ x

2

− 2dx cos(π − 2β − α) = d

2

+ x

2

+ 2dx cos(2β + α),

c

2

= d

2

+ y

2

− 2dy cos(π − 2α − β) = d

2

+ y

2

+ 2dy cos(2α + β),

z

2

= d

2

+ x

2

− 2dx cos β,

z

2

= d

2

+ y

2

− 2dy cos α,

d

2

+ x

2

+ 2dx cos(2β + α) = d

2

+ y

2

+ 2dy cos(2α + β),

d

2

+ x

2

− 2dx cos β = d

2

+ y

2

− 2dy cos α,

4dx(cos(2β + α) + cos β) = 4dy(cos(2α + β) + cos α),

x

cos

3β + α

2 cos

α

+ β

2 = y cos

3α + β

2 cos

α

+ β

2 .

ø Å|ÓDÌ"Å|âqÊ-¾¯ØDâqÁ

0 <

α

+ β

2 <

π

4

¿¯Ñ7Å|½

0 <

3α + β

2 <

π

2

Ä

0 <

3β + α

2 <

π

2 .

¿ ½jÅ¯Í&Å¯Ò!ÃEÄÊ-¾¯Ø!âqÁ

3α + β

2 <

π

2

Ä

3β + α

2 <

π

2 .

l\ÅwÈ×EÄÁqÑ&ÌÎ½jÅ|ÓDÌ"Å|âqÊ-¾¯ØDâqÁ

3α + β

2 =

α

+ β

2 + α <

π

4 +

π

2 =

3

4 π.

û ¿!Ò!¿¯ÀEÃEÄÁ

3β + α

2 <

3

4 π.

ìÒ!¾DÀD¾+ÃE×:æ

π

2 <

3α + β

2 <

3

4 π

Ä

3β + α

2

6 π

2 ,

Ë&¿¶Ì"Å|Ñ&ÓEÃEÁqÇ

x

cos

3β + α

2 cos

α

+ β

2 = y cos

3α + β

2 cos

α

+ β

2

Ñ&àwÌLÃE¿wÌ"Å|âqÃD¾)Ì"Å|Ñ&ÓEÃEÇ¯¿wÌLÄ

x

cos

3β + α

2 = y cos

3α + β

2

ÃEÄÁªÊAà¯Þ!Ï¹ÀD¾¶½jÅwÈÐ&ã|æbÜ5Ë7Ô¯Ò

3α + β

2

6 π

2

Ä

3β + α

2

6 π

2 .

(6)

ìÒ!¾DÀD¾

3α + β

2 =

π

2

Ä

3β + α

2 =

π

2 ,

Ë&¿

α

+ β = π.

ëÂ`½q¾DÌLÄµÐ&Â`ÄÁ×EÑ&½q¾D×bÅ¯Ò!ÁqÇ

3α + β

2 =

π

2

Ä

3β + α

2 <

π

2

ÆÓEÀ

3α + β

2 <

π

2

Ä

3β + α

2 =

π

2 ,

½jÅwÈÐ&ãÃEÄÁªÊA¿¯âqÁ¯Ø!Àp¿A×EÑ&½qÁjÂ`½q¾¶Ë&ÁqÊ-Ó6Ì"Å|Ñ&ÓEÃEÁqÇ

x

cos

3β + α

2 = y cos

3α + β

2 .

Ü5Ë7Ô¯Ò

0 <

3α + β

2 <

π

2

Ä

0 <

3β + α

2 <

π

2 .

ìÒ!¾DÀD¾Ë&ÁqÑ7Å|½

α > β

Ø!Ë&¿

x > y

¿¯Ñ7Å|½

3α + β

2 >

3β + α

2 .

Ü5Ë7Ô¯Ò

cos

3α + β

2 <

cos

3β + α

2

Ä

y

cos

3α + β

2 < x

cos

3β + α

2 .

ëBË&Ñ&½q¾DÊ¶Å|ÆÄµÐÊ-¾Í×EÑ&½qÁjÂ`½qÃE¿5Ð&ã\½Ì"Å|Ñ&ÓEÃEÇDÄÁqÊ

x

cos

3β + α

2 = y cos

3α + β

2 .

û ¿!Ò!¿¯ÀEÃEÄÁÌL¾DÇDÆÓbÂ`½jÅ|Ê-¾×EÑ&½q¾D×bÅ¯Ò!ÁqÇ

α < β.

ø ÅwË&ÁqÊ

α

= β.

ÁÂ¹^¾ÀG·ÈsÃ û Ñ&½q¾jÈÊAÄÈÊ-¾+¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅBÈÅ|Ç+ÃbÅ)Ñ&¾!ÍÓEÃEÇDÓüï¯ØEÅAÌLÄåjÂ

|AD| = |BE|.

a

= b.

2α > 2β.

ç Ë&ÁjÒ!¾

sin α > sin β

Ä:×p¿¯ÃEÄÁqÌ"Å|â

P

M

ABD

=

1

2 · dc · sin α,

P

M

ABE

=

1

2 · dc · sin β,

Ë&¿

P

M

ABD

> P

M

ABE

,

½jÅwË&ÁqÊ

P

M

ADC

< P

M

BEC

.

Ü5Ë7Ô¯Ò/×p¿¯ÃEÄÁqÌ"Å|â

P

MADC

=

1

2 · bd · sin α

Ä

P

MBEC

=

1

2 · da · sin β,

Ë&¿

1

2 · bd · sin α <

1

2 · sin β,

b

sin β

<

a

sin α

.

ÜDÇ¯¿¯Ñ&¿

0 < β < α <

π

2

,

Ë&¿

cos β > cos α > 0

Ä

0 <

1 2 cos β

<

1 2 cos α

.

(7)

° Ü5Ë7Ô¯Ò/¿¯Ñ7Å|½½ÃEÄÁqÑ&àwÌLÃE¿5Ð&Â`Ä

b

sin

β

<

a

sin

α

Ò!¿5ÍË7ÅwÈÁqÊ-¾Ç¯¿¯ÆÁÈÃE¿9q

b

2 sin β cos β

<

a

2 sin α cos α

,

b

sin 2β

<

a

sin 2α

.

ëBË&Ñ&½q¾DÊ¶Å|ÆÄµÐÊ-¾Í×EÑ&½qÁjÂ`½qÃE¿5Ð&ã)ß½ËÌLÄÁqÑ7Ò!½qÁqÃEÄÁqÊýÍÄÃDÓbÍàwÌÎÒ!ÆµÅ)Ë&Ñ&à|ÈÇwÔwË7Å

ABC

á¢æ mBÃbÅ|Æ¿¯Þ¯ÄµÂ`½qÃEÄÁ\ÊA¿¯âqÃbÅ$ÓE½jÅ¯Í&Å¯Ò!ÃEÄµã|Ø5âqÁ×EÑ&½q¾D×EÓbÍ½jÂ`½qÁqÃEÄÁ

2β > 2α

×EÑ&¿wÌ"Å¯Ò!½qÄpÒ!¿AÍ×EÑ&½qÁ`Õ Â`½qÃE¿5Ð&Â`Äæ ø ÅwË&ÁqÊ

2β = 2α

Ø

a

= b.

ÁÂ¹^¾ÀGÊÈsÃ û Ñ&½q¾jÈÊAÄÈÊ-¾+¿¯½qÃbÅ¯Â`½qÁqÃEÄµÅBÈÅ|Ç+ÃbÅ)Ñ&¾!ÍÓEÃEÇDÓüï¯ØEÅAÌLÄåjÂ

|AD| = |BE|.

a

= b.

¶ Å|Ê-¾

|AD| =

2bc

b

+ c

cos α,

|BE| =

2ac

a

+ c

cos β.

a > b.

ç Ë&ÁjÒ!¾

α > β

Ä

0 < β < α <

π

2 .

Ü5Ë7Ô¯Ò

cos β > cos α.

û ¿¯ÃbÅ¯ÒDË&¿

2bc

b

+ c

cos α =

2ac

a

+ c

cos β,

b

· (a + c) cos α = a · (b + c) cos β,

ab

· (cos α − cos β) = ac · cos β − bc · cos α.

Ë

ÁqÌ"ÅüÍË&Ñ&¿¯ÃbÅ¿5ÍË7ÅwË&ÃEÄÁÈ)Ñ&àwÌLÃE¿5Ð&Â`Ä:ÈÁjÍË¶ÓwÈÁqÊAÃbÅEØZ½jÅ¯Ð)×EÑ7ÅÖÌ"ÅÒ!¿!ÒEÅwË&ÃEÄµÅEØPÞ5Ò!¾Dâ

ac > bc

Ä

cos β > cos α > 0.

ëBË&Ñ&½q¾DÊ¶Å|ÆÄµÐÊ-¾Í×EÑ&½qÁjÂ`½qÃE¿5Ð&ã|æ mBÃbÅ|Æ¿¯Þ¯ÄµÂ`½qÃEÄÁªÊA¿¯âqÃbÅ¶ÓE½jÅ¯Í&Å¯Ò!ÃEÄµã|ØbâqÁ$×EÑ&½q¾D×EÓbÍ½jÂ`½qÁqÃEÄÁ

a < b

×EÑ&¿wÌ"Å¯Ò!½qÄÒ!¿+Í×EÑ&½qÁjÂ`½`Õ ÃE¿5Ð&Â`Äæ ø ÅwË&ÁqÊ

a

= b.

l\Åª½jÅ|Ç¯¿¯ÚbÂ`½qÁqÃEÄÁ"×p¿!ÒEÅ|Ê-¾ ¹^¾ÀGÈÂ·

ø ÅDÏ¹à¯âqÊ-¾¯Ø!âqÁ$ÌË&Ñ&à|ÈÇwÔ¯Â`ÄÁ

ABC

×EÑ&½q¾!ÍË7ÅwÈÔ¯Â`ÁªÍ&Ô¶¿!ÒEÂ`ÄÃEÇDÄ

AD

Ä

BE

Ò!ÌLÓbÍÄÁjÂ`½qÃD¾!Â7É Ò!Ì1à!Â7É-ÈÁqÞ¯¿AÇwÔwË&àwÌßÑ&¾!ÍqæÎ5á¢æ

D

0

A

B

C

D

E

α

β

§¨^©|ª¤«7¬Ï^¯

(8)

û Ñ&½q¾D×EÓbÐ&ã`Ê-¾¯ØEâqÁ

|BC| > |AC|.

Ü5Ë7Ô¯Ò

|^BAD| > |^EBA|.

"ÒEÅ|ÆÁÈL½ËÌLÄÁqÑ7Ò!½qÁqÃEÄµÅ Â`¿5ÍÄÃDÓbÍàwÌ'q

|BD| > |AE|.

ø ÅwË&ÁqÊ

|^ADB| > |^AEB|.

û Ñ&½qÁq½¶×EÓEÃEÇ5Ë

A

×EÑ&¿wÌ"Å¯Ò!½qÄÕ Ê-¾/×EÑ&¿5ÍË7ÔAÑ&àwÌLÃE¿¯ÆÁqÞ!ÏÔAÒ!¿

BC.

lBÄÁjÂ7É×EÓEÃEÇ5Ë

D

0

ÀpåjÒ!½qÄÁ$Ë7Å|ÇDÄÊW×EÓEÃEÇ5Ë&ÁqÊWË&ÁÈB×EÑ&¿5ÍË&ÁÈ`Ø âqÁ

|AD

0

_{| = |BD|}

¿¯Ñ7Å|½

E

Ä

D

0

ÆÁqâjÔ?×p¿ù×EÑ&½qÁjÂ`ÄÌLÃD¾!Â7ÉOÍË&Ñ&¿¯ÃbÅ¯Â7ÉT×EÑ&¿5ÍË&ÁÈ

AB

ß½q¿¯À:æ Ñ&¾!ÍÓEÃEÁqÇÎ5á¢æ ç Ë&ÁjÒ!¾

|^DBA| = |^BAD

0

_|

Ä1Ë&Ñ&à|ÈÇwÔwË¾

ABD

Ä

ABD

0

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