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Theorem cevath 27007
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 27006 three times and then using cevathlem1 27005 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 27002. (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 968 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 cevath.c . . . . . 6  |-  ( ph  ->  O  e.  CC )
53, 4subcld 9202 . . . . 5  |-  ( ph  ->  ( B  -  O
)  e.  CC )
62simp3d 969 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76, 4subcld 9202 . . . . 5  |-  ( ph  ->  ( C  -  O
)  e.  CC )
85, 7jca 518 . . . 4  |-  ( ph  ->  ( ( B  -  O )  e.  CC  /\  ( C  -  O
)  e.  CC ) )
91, 8sigarimcd 27000 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  e.  CC )
102simp1d 967 . . . 4  |-  ( ph  ->  A  e.  CC )
11 cevath.b . . . . 5  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
1211simp1d 967 . . . 4  |-  ( ph  ->  F  e.  CC )
1310, 12subcld 9202 . . 3  |-  ( ph  ->  ( A  -  F
)  e.  CC )
1410, 4subcld 9202 . . . . 5  |-  ( ph  ->  ( A  -  O
)  e.  CC )
157, 14jca 518 . . . 4  |-  ( ph  ->  ( ( C  -  O )  e.  CC  /\  ( A  -  O
)  e.  CC ) )
161, 15sigarimcd 27000 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  e.  CC )
179, 13, 163jca 1132 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  e.  CC  /\  ( A  -  F
)  e.  CC  /\  ( ( C  -  O ) G ( A  -  O ) )  e.  CC ) )
1812, 3subcld 9202 . . 3  |-  ( ph  ->  ( F  -  B
)  e.  CC )
1914, 5jca 518 . . . 4  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
201, 19sigarimcd 27000 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
2111simp3d 969 . . . 4  |-  ( ph  ->  E  e.  CC )
226, 21subcld 9202 . . 3  |-  ( ph  ->  ( C  -  E
)  e.  CC )
2318, 20, 223jca 1132 . 2  |-  ( ph  ->  ( ( F  -  B )  e.  CC  /\  ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  E
)  e.  CC ) )
2421, 10subcld 9202 . . 3  |-  ( ph  ->  ( E  -  A
)  e.  CC )
2511simp2d 968 . . . 4  |-  ( ph  ->  D  e.  CC )
263, 25subcld 9202 . . 3  |-  ( ph  ->  ( B  -  D
)  e.  CC )
2725, 6subcld 9202 . . 3  |-  ( ph  ->  ( D  -  C
)  e.  CC )
2824, 26, 273jca 1132 . 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 968 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 )
3129simp1d 967 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 )
3229simp3d 969 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 )
3330, 31, 323jca 1132 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
346, 10, 33jca 1132 . . . 4  |-  ( ph  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
3521, 12, 253jca 1132 . . . 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 969 . . . . 5  |-  ( ph  ->  ( ( C  -  O ) G ( F  -  O ) )  =  0 )
3836simp1d 967 . . . . 5  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
3936simp2d 968 . . . . 5  |-  ( ph  ->  ( ( B  -  O ) G ( E  -  O ) )  =  0 )
4037, 38, 393jca 1132 . . . 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 969 . . . . 5  |-  ( ph  ->  ( ( C  -  E ) G ( A  -  E ) )  =  0 )
4341simp1d 967 . . . . 5  |-  ( ph  ->  ( ( A  -  F ) G ( B  -  F ) )  =  0 )
4441simp2d 968 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
4542, 43, 443jca 1132 . . . 4  |-  ( ph  ->  ( ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0 ) )
4632, 31, 303jca 1132 . . . 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 27006 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  x.  ( A  -  F )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( F  -  B ) ) )
483, 6, 103jca 1132 . . . 4  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
4925, 21, 123jca 1132 . . . 4  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
5039, 37, 383jca 1132 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0 ) )
5144, 42, 433jca 1132 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0 ) )
5230, 32, 313jca 1132 . . . 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 27006 . . 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 27006 . . 3  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
5547, 53, 543jca 1132 . 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 27005 1  |-  ( ph  ->  ( ( ( A  -  F )  x.  ( C  -  E
) )  x.  ( B  -  D )
)  =  ( ( ( F  -  B
)  x.  ( E  -  A ) )  x.  ( D  -  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   CCcc 8780   0cc0 8782    x. cmul 8787    - cmin 9082   *ccj 11628   Imcim 11630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-2 9849  df-cj 11631  df-re 11632  df-im 11633
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