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Theorem cevath 32247
Description: Ceva's theorem. Let  A B C be a triangle and let points  F,  D and  E lie on sides  A B,  B C,  C A correspondingly. Suppose that cevians  A D,  B E and  C F intersect at one point  O. Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 32246 three times and then using cevathlem1 32245 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function  G as a collinearity indicator. For justification of that use, see sigarcol 32242. This is Metamath 100 proof #61. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Hypotheses
Ref Expression
cevath.sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
cevath.a  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
cevath.b  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
cevath.c  |-  ( ph  ->  O  e.  CC )
cevath.d  |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O
) )  =  0  /\  ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0 ) )
cevath.e  |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0 ) )
cevath.f  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
Assertion
Ref Expression
cevath  |-  ( ph  ->  ( ( ( A  -  F )  x.  ( C  -  E
) )  x.  ( B  -  D )
)  =  ( ( ( F  -  B
)  x.  ( E  -  A ) )  x.  ( D  -  C ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y   
x, O, y    x, E, y    x, F, y
Allowed substitution hints:    ph( x, y)    G( x, y)

Proof of Theorem cevath
StepHypRef Expression
1 cevath.sigar . . . 4  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
2 cevath.a . . . . . . 7  |-  ( ph  ->  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )
)
32simp2d 1009 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 cevath.c . . . . . 6  |-  ( ph  ->  O  e.  CC )
53, 4subcld 9950 . . . . 5  |-  ( ph  ->  ( B  -  O
)  e.  CC )
62simp3d 1010 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76, 4subcld 9950 . . . . 5  |-  ( ph  ->  ( C  -  O
)  e.  CC )
85, 7jca 532 . . . 4  |-  ( ph  ->  ( ( B  -  O )  e.  CC  /\  ( C  -  O
)  e.  CC ) )
91, 8sigarimcd 32240 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  e.  CC )
102simp1d 1008 . . . 4  |-  ( ph  ->  A  e.  CC )
11 cevath.b . . . . 5  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
1211simp1d 1008 . . . 4  |-  ( ph  ->  F  e.  CC )
1310, 12subcld 9950 . . 3  |-  ( ph  ->  ( A  -  F
)  e.  CC )
1410, 4subcld 9950 . . . . 5  |-  ( ph  ->  ( A  -  O
)  e.  CC )
157, 14jca 532 . . . 4  |-  ( ph  ->  ( ( C  -  O )  e.  CC  /\  ( A  -  O
)  e.  CC ) )
161, 15sigarimcd 32240 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  e.  CC )
179, 13, 163jca 1176 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  e.  CC  /\  ( A  -  F
)  e.  CC  /\  ( ( C  -  O ) G ( A  -  O ) )  e.  CC ) )
1812, 3subcld 9950 . . 3  |-  ( ph  ->  ( F  -  B
)  e.  CC )
1914, 5jca 532 . . . 4  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
201, 19sigarimcd 32240 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
2111simp3d 1010 . . . 4  |-  ( ph  ->  E  e.  CC )
226, 21subcld 9950 . . 3  |-  ( ph  ->  ( C  -  E
)  e.  CC )
2318, 20, 223jca 1176 . 2  |-  ( ph  ->  ( ( F  -  B )  e.  CC  /\  ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  E
)  e.  CC ) )
2421, 10subcld 9950 . . 3  |-  ( ph  ->  ( E  -  A
)  e.  CC )
2511simp2d 1009 . . . 4  |-  ( ph  ->  D  e.  CC )
263, 25subcld 9950 . . 3  |-  ( ph  ->  ( B  -  D
)  e.  CC )
2725, 6subcld 9950 . . 3  |-  ( ph  ->  ( D  -  C
)  e.  CC )
2824, 26, 273jca 1176 . 2  |-  ( ph  ->  ( ( E  -  A )  e.  CC  /\  ( B  -  D
)  e.  CC  /\  ( D  -  C
)  e.  CC ) )
29 cevath.f . . . 4  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
3029simp2d 1009 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 )
3129simp1d 1008 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 )
3229simp3d 1010 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 )
3330, 31, 323jca 1176 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
346, 10, 33jca 1176 . . . 4  |-  ( ph  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
3521, 12, 253jca 1176 . . . 4  |-  ( ph  ->  ( E  e.  CC  /\  F  e.  CC  /\  D  e.  CC )
)
36 cevath.d . . . . . 6  |-  ( ph  ->  ( ( ( A  -  O ) G ( D  -  O
) )  =  0  /\  ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0 ) )
3736simp3d 1010 . . . . 5  |-  ( ph  ->  ( ( C  -  O ) G ( F  -  O ) )  =  0 )
3836simp1d 1008 . . . . 5  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
3936simp2d 1009 . . . . 5  |-  ( ph  ->  ( ( B  -  O ) G ( E  -  O ) )  =  0 )
4037, 38, 393jca 1176 . . . 4  |-  ( ph  ->  ( ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0  /\  ( ( B  -  O ) G ( E  -  O
) )  =  0 ) )
41 cevath.e . . . . . 6  |-  ( ph  ->  ( ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0 ) )
4241simp3d 1010 . . . . 5  |-  ( ph  ->  ( ( C  -  E ) G ( A  -  E ) )  =  0 )
4341simp1d 1008 . . . . 5  |-  ( ph  ->  ( ( A  -  F ) G ( B  -  F ) )  =  0 )
4441simp2d 1009 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
4542, 43, 443jca 1176 . . . 4  |-  ( ph  ->  ( ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0 ) )
4632, 31, 303jca 1176 . . . 4  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 ) )
471, 34, 35, 4, 40, 45, 46cevathlem2 32246 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  x.  ( A  -  F )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( F  -  B ) ) )
483, 6, 103jca 1176 . . . 4  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
4925, 21, 123jca 1176 . . . 4  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
5039, 37, 383jca 1176 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0 ) )
5144, 42, 433jca 1176 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0 ) )
5230, 32, 313jca 1176 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 ) )
531, 48, 49, 4, 50, 51, 52cevathlem2 32246 . . 3  |-  ( ph  ->  ( ( ( A  -  O ) G ( B  -  O
) )  x.  ( C  -  E )
)  =  ( ( ( B  -  O
) G ( C  -  O ) )  x.  ( E  -  A ) ) )
541, 2, 11, 4, 36, 41, 29cevathlem2 32246 . . 3  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
5547, 53, 543jca 1176 . 2  |-  ( ph  ->  ( ( ( ( B  -  O ) G ( C  -  O ) )  x.  ( A  -  F
) )  =  ( ( ( C  -  O ) G ( A  -  O ) )  x.  ( F  -  B ) )  /\  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( C  -  E
) )  =  ( ( ( B  -  O ) G ( C  -  O ) )  x.  ( E  -  A ) )  /\  ( ( ( C  -  O ) G ( A  -  O ) )  x.  ( B  -  D
) )  =  ( ( ( A  -  O ) G ( B  -  O ) )  x.  ( D  -  C ) ) ) )
5617, 23, 28, 33, 55cevathlem1 32245 1  |-  ( ph  ->  ( ( ( A  -  F )  x.  ( C  -  E
) )  x.  ( B  -  D )
)  =  ( ( ( F  -  B
)  x.  ( E  -  A ) )  x.  ( D  -  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   CCcc 9507   0cc0 9509    x. cmul 9514    - cmin 9824   *ccj 12940   Imcim 12942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-cj 12943  df-re 12944  df-im 12945
This theorem is referenced by: (None)
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