MGNE_U2_contenido - Wendy Flores

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 
 
 
 
 
 
 
 
 
 
 
 1
2 3 l ∥l 1 2 31 
 
1 2 3 4 l1 ∥l 1 3 1 4 3 4 l ∥l4
 
 
t ' t ' t
B 'B ' A C
* *A B C 'A 'C ' '* '* 'A B C
ΔAʹBʹB, ΔABBʹ, ΔCʹBʹB, ΔCBBʹ
∢𝐴𝐵𝐵ʹ, ∢𝐶ʹBʹB ∢𝐴ʹBʹB, ∢𝐶𝐵𝐵ʹ
 
 
t ' , , , ', ', 'A B C A B C
∢𝐴ʹBʹB ≅ ∢𝐶𝐵𝐵ʹ 
 
 '
D ' D t
C 'C
 D
 E ' 'B A 'B E BD
E 'B E BD
 'BB
' 'BB BD B E '' BDB B E B
 
ΔBʹBD ΔBBʹE
 ∢𝐷𝐵ʹB ∢𝐸𝐵ʹB ∢𝐸𝐵𝐵ʹ
∢𝐷𝐵𝐵ʹ
 E
  '
 
 D l ∥ l 
∎ 
Corolario 2.7.1. 
  t
l ∥l´ ∎ 
 
Corolario 2.7.2. P
m P
 t
P m t P m
t l ∥ m.
m
P
 
 
 
 
 
 
P
P
 
 
 P
P
 P
P
 Q PQ
PQ
 m P PQ m
 
m PQ
 R Q t R
t R
 S t PS t
PS t
 PS t 𝑃�̂� ∥ l
 PS m S
m S m
 ▭
 S m
∎ 
 
Corolario 2.8.1. P
P
R
R
∎
 
 
∢𝐴𝐵𝐶 ∢𝐷𝐸𝐹 ABC ∢𝐷𝐸𝐹
∢𝐴𝐵𝐶 < ∢𝐷𝐸𝐹 EH ED EF
∢𝐴𝐵𝐶 ≅ ∢𝐷𝐸𝐻
∢𝑪𝑩𝑨 ∢𝑫𝑬𝑭
 
 , ,A B C ∢𝐵𝐴𝐶
 
∢𝑩𝑨𝑪 𝚫𝐀𝐁𝐂
 D BC * *B C D
D * *B C D
 ∢𝐵𝐴𝐶 ≅ ∢𝐴𝐶𝐷 BA BC
B
 ∢𝐵𝐴𝐶 > ∢𝐴𝐶𝐷 AE ∢𝐴𝐶𝐷 ≅ ∢𝐶𝐴𝐷
AE ∢𝑨𝑪𝑫 ≅ ∢𝑪𝑨𝑫
 G AE * *B G C
 
G * *B G C
(vi). AE CD A
 
(vii). ∢𝐵𝐴𝐶 < ∢𝐴𝐶𝐷 
 ∢𝐴𝐵𝐶 < ∢𝐴𝐶𝐷 F AC
* *A C F
AC * *A C F
 ∢𝐴𝐶𝐷 ≅ ∢𝐹𝐵𝐶 C
 ∢𝐴𝐵𝐶 < ∢𝐵𝐶𝐹
 ∢𝐴𝐵𝐶 < ∢𝐴𝐷𝐹
∎ 
Proposición 2.1. ΔABC ΔDEF AC DF ∢𝐴 ≅ ∢𝐷 ∢𝐵 ≅
∢𝐸
Proposición 2.2. ΔABC ≅ ΔDEF 
 
Proposición 2.3. 
Proposición 2.4. 
Proposición 2.5. 
Proposición 2.6. 
Proposición 2.7. 
 
Proposición 2.8. ΔABC ΔDEF AB DE BC EF
∢𝐵 < ∢𝐸 AC DF
Proposición 2.9. 
 A
 A
  0 180A 
    A B BA 
 AC DAB      DAC CAB DAB 
   90A  A
 
  0,180x A  A x
 A B     180A B 
    A B A B
A A   90A 
A   90A 
Corolario 2.10.1. 
180
 OP
AB AB
 0AB  1OP 
 AB CD AB CD
 * *A B C AB BC AC 
 AB CD AB CD
 x 0x  AB AB x
Corolario 2.11.1. A B C
AC AB BC 
 A B C
 
A B C
 BA
D * *A B D BD BC
D * *A B D BD BC
 CBD BBCD DC
CBD
 AB BD AD 
 BC BD AB BC AD 
 CB CA CD
∢𝐵𝐶𝐷 < ∢𝐴𝐶𝐷
 ∢𝐵𝐶𝐷 < ∢𝐴𝐶𝐷 ∢𝐵𝐶𝐷 ≅ ∢𝐵𝐷𝐶 ∢𝐴𝐷𝐶 < ∢𝐴𝐶𝐷
 
∢𝑨𝑫𝑪 ∢𝑨𝑪𝑫
 AD AC
 AD AC
 AB BD AD AC  
∎ 
 ΔABC D BC E
AD * *A D E AD DE
ΔABC
ΔAEC AEC EAC
 
1
2
BAC
 ΔABC D BC E
AD * *A D E AD DE
D AE BC
 
 D CBDA DE
BD CD ED AD
 BDA CDE
𝚫𝐁𝐃𝐀 𝚫𝐂𝐃𝐄
 DEC DAB BAC BAD DAC CEA EAC   
DEC DAB
 EABD CD ECA ECD DCA ECB BCA   
ABD ECD
 
 
             
     
BAC ACB CBA CEA EAC ACB ECB
CEA EAC ACE
      
 
  
           2min ,BAC CEA EAC CEA EAC 
      1in
2
,m CEA EAC BAC
 
180
 
ABC 180
 0p        180A B C p   
 
1ABC
ABC
1ABC      1 180A B C p       1
1
2
AB BACC 
1ABC
 
1ABC
2AB C
     2 180A B C p         12
1 1
2 4
B AC BACA CB  
 
2AB C
 k
kAB C
      180kA B C p       
1
2k
kA BACB C 
 
0k   
0
1
2
k
BAC p
 
0k
AB C
         
0
180 kp A B C A p C      
    180 A C 
       180A B C  
Corolario 2.12.1. 
ABCD
AB CD CD
AB AB
CD
 
Corolario 2.12.2. 
360
ΔABC ABC
      180A B C ABC   
180
180
 ΔABC D * *A D B
ABC ACD DCB   
 CD ACB
CD ACB
      ACD DCB ACB 
 
 ADC CDB
    180ADC CDB 
 
     
     
     
180
180
180
ABC BCA CAB
DBC BCD C
ABC
DBCDB
ADC DCA C CAD AD



   
   
   
 
ABC ACD DCB   
Corolario 2.13.1. 
ABC 180
ACD DCB 180
ACD
DCB 180 0ACD DCB  
0 0 0ABC ACD DCB      
ABC 180 0ABC 
0ACD DCB ABC     ACD DCB
0ACD DCB  
ΔACD ΔDCB 180
 180
180
Corolario 2.14.1. 
 
180
 
Corolario 2.15.1. 
360
 ABCD
 AC ΔABC ΔACD
𝚫𝐀𝐁𝐂 𝚫𝐀𝐂𝐃
 
180
 ABCD AC AB
AD
AC AB AD
 
     BAC CAD BAD 
      ACB ACD BCD 
 
ABCD 180
 
 
 
 
 
ΔABC ΔDEF ΔABC
ΔDEF
 
ΔABC ΔDEF ABC DEF
 ΔABC DE
ΔDEF ABC DEF
 
 P
P
 Q PQ m P
m PQ
 
PQ
 PQ m PQ m m
 n m P
n P
 
1n n P PQ 1m
1m m
1n n P
 R
1n S PQ
PS QR
 
SR PQ
 ΔPSR PQ
T ΔPSR ∥ ΔPQT P P S Q R T
T R PQ
T R PQ
1T
R PQ
1PQT PQT
PQT
1PQT
 RTPQ PS
 PQ PS
PR PT T PR
 PPQT SR T
n n T
T n
 
 m
 
 ABC ' ' 'A B C
'A A 'B B 'C C
 
ABC
' ' 'A B C
  , ,AB AC BC  ' ', ' ', ' 'A B A C B C
ABC ' ' 'A B C
 
 
' 'AB A B ' 'AC A C
 ''B ''C AB AC
' '''AB A B ' '''AC A C
 '' ''AB C ' ' 'A B C
'' ''AB C B '' ''AC B C
 ' ' 'ABC A B C
'' ''AB C B '' ''AC B C
 '' ''BC B C
'' ''BB C C
 
       '' '' '' '' 180B BB C C CC B   
 '' ''BB C C
360
 
 
' , , ,A B C 
', ', ',A B C  '
, , ,A B C 
' , , ,A B C 
 ' '' BB CAA C  
 
 
 AB
AB
 m n AB A B
m n AB
 C D m n
C D AB AC BD
C D AC BD
 CD ABCD
 
ABCD
 
 
 
 
 ABCD A B
AC DB DC 
 ABCD A B
AC DB
ABCD
 AD BC
CAB DBA BACB DA CB DA
CAB DBA
 DCBD AC BCD ADC
 C ACB BCD  D BDA ADC 
 
 
     
   
 
C ACB BCD
BDA ADB
D
 
 

 DC 
 
  ,A B ' 'A B 
' 'ABB A
M 'M AB ' 'A B
' 'ABB A
 
 BA 
''A AM B BM 'A M 'B M
'A AM 'B BM
 '' ' 'A M M B M M
' 'A M M ' 'B M M
'' 'M M A B
' 'A M M ' 'B M M
 
' ' ' 'A MM B MM ''A MA B MB
 ' ' ' 'AMA A MM AMM  ' ' ' 'BMB B MM BMM 
'A MA 'B MB
 
   ' 'BA MA MB
'A MA 'B MB
'A MA 'B MB
 ' 'A M MA
' 'AA MM
 

 
 , ,A B C
'
, ,A B C
 ' 'A B BA ' 'A C CA ' 'B C CB
' 'A B BA ' 'A C CA ' 'B C CB
 ''A AB B BA ''A AC C CA ''B BC C CB
 
'B BA 'B BC
 
'B BA 'B BC
 ' 'A B BA ' 'A C CA ' 'B C CB
 A B C
 
A B  
  A B

A B
 
 'A 'B  ' 'A B BA
'A 'B '
A B
'
' 'A B BA
 M 'M AB ' 'A B
 'M M 
'M M 
 
'MM 'MM
 
A B M
AB A B 
  'MM

'MM 
 'MM 
 A B M
AB
 'A 'B 
A B
'MM
 
 '' MAM M B M '' MAM M B M
' 'AM BM
'AM M 'BM M
 
       ' ' 90 ' 90 ' ' 'A M A AM M BM M B M B    
 '' ' 'A M A B M B ' ' ' 'AA M BB M
'AA 'BB
 
 
P PQ
P m PQ P
m PQ m
n m P
n
 kPR P
PR
kPR m PR
kPR m PR
PR
kPR
 
 P Q
PQ 
PX 'PX
PQ P
PX 'PX
PQ
'PXPQ X Q
 P
PQ P m
PQ P m
m
 S m P SQ 1
T SQ
PT
2 1
1Q 2S
 
1 2
 T SQ
1
TQ
1  21, 
X SQ
21 * *P X P 1 1P
2 2P  1 2
PX
PX
 21, 
 PX
PX U
 
PX U
 V * *V U Q U V
SQ P
V SQ VP
SQ Y * *Y X Q
2Y
PY PX
PX
 
XPQ 'X PQ
   'XPQ X PQ
'PR 'PX PQ
' XR PQ PQ
 
 R
'R PQ * * 'R Q R 'RQ R Q
 'PRPQ R Q 'PRPQ R Q
XRPQ PQ
PR PX PQ
'X X
XPQ 'X PQ PX
'PX  PQ
  90PQ    90PQ 
PQ
0 90
OI OI   45OI 
 m
m
 
 
m m
 m m m
m
 
 
 
 
http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&field-author=Richard%20Courant&search-alias=books&sort=relevancerank
http://www.amazon.com/s/ref=ntt_athr_dp_sr_2?_encoding=UTF8&field-author=Herbert%20Robbins&search-alias=books&sort=relevancerank
http://www.amazon.com/s/ref=ntt_athr_dp_sr_3?_encoding=UTF8&field-author=Ian%20Stewart&search-alias=books&sort=relevancerank
http://en.wikipedia.org/wiki/Elliptic_geometry
http://www.math.brown.edu/~banchoff/gc/elliptic/elliptic.html
http://web.mnstate.edu/peil/geometry/C2EuclidNonEuclid/7elliptic.htm