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Theorem cevath 29903
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 29902 three times and then using cevathlem1 29901 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 29898. 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 1001 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 cevath.c . . . . . 6  |-  ( ph  ->  O  e.  CC )
53, 4subcld 9718 . . . . 5  |-  ( ph  ->  ( B  -  O
)  e.  CC )
62simp3d 1002 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76, 4subcld 9718 . . . . 5  |-  ( ph  ->  ( C  -  O
)  e.  CC )
85, 7jca 532 . . . 4  |-  ( ph  ->  ( ( B  -  O )  e.  CC  /\  ( C  -  O
)  e.  CC ) )
91, 8sigarimcd 29896 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  e.  CC )
102simp1d 1000 . . . 4  |-  ( ph  ->  A  e.  CC )
11 cevath.b . . . . 5  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
1211simp1d 1000 . . . 4  |-  ( ph  ->  F  e.  CC )
1310, 12subcld 9718 . . 3  |-  ( ph  ->  ( A  -  F
)  e.  CC )
1410, 4subcld 9718 . . . . 5  |-  ( ph  ->  ( A  -  O
)  e.  CC )
157, 14jca 532 . . . 4  |-  ( ph  ->  ( ( C  -  O )  e.  CC  /\  ( A  -  O
)  e.  CC ) )
161, 15sigarimcd 29896 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  e.  CC )
179, 13, 163jca 1168 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  e.  CC  /\  ( A  -  F
)  e.  CC  /\  ( ( C  -  O ) G ( A  -  O ) )  e.  CC ) )
1812, 3subcld 9718 . . 3  |-  ( ph  ->  ( F  -  B
)  e.  CC )
1914, 5jca 532 . . . 4  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
201, 19sigarimcd 29896 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
2111simp3d 1002 . . . 4  |-  ( ph  ->  E  e.  CC )
226, 21subcld 9718 . . 3  |-  ( ph  ->  ( C  -  E
)  e.  CC )
2318, 20, 223jca 1168 . 2  |-  ( ph  ->  ( ( F  -  B )  e.  CC  /\  ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  E
)  e.  CC ) )
2421, 10subcld 9718 . . 3  |-  ( ph  ->  ( E  -  A
)  e.  CC )
2511simp2d 1001 . . . 4  |-  ( ph  ->  D  e.  CC )
263, 25subcld 9718 . . 3  |-  ( ph  ->  ( B  -  D
)  e.  CC )
2725, 6subcld 9718 . . 3  |-  ( ph  ->  ( D  -  C
)  e.  CC )
2824, 26, 273jca 1168 . 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 1001 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 )
3129simp1d 1000 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 )
3229simp3d 1002 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 )
3330, 31, 323jca 1168 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
346, 10, 33jca 1168 . . . 4  |-  ( ph  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
3521, 12, 253jca 1168 . . . 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 1002 . . . . 5  |-  ( ph  ->  ( ( C  -  O ) G ( F  -  O ) )  =  0 )
3836simp1d 1000 . . . . 5  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
3936simp2d 1001 . . . . 5  |-  ( ph  ->  ( ( B  -  O ) G ( E  -  O ) )  =  0 )
4037, 38, 393jca 1168 . . . 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 1002 . . . . 5  |-  ( ph  ->  ( ( C  -  E ) G ( A  -  E ) )  =  0 )
4341simp1d 1000 . . . . 5  |-  ( ph  ->  ( ( A  -  F ) G ( B  -  F ) )  =  0 )
4441simp2d 1001 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
4542, 43, 443jca 1168 . . . 4  |-  ( ph  ->  ( ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0 ) )
4632, 31, 303jca 1168 . . . 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 29902 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  x.  ( A  -  F )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( F  -  B ) ) )
483, 6, 103jca 1168 . . . 4  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
4925, 21, 123jca 1168 . . . 4  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
5039, 37, 383jca 1168 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0 ) )
5144, 42, 433jca 1168 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0 ) )
5230, 32, 313jca 1168 . . . 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 29902 . . 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 29902 . . 3  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
5547, 53, 543jca 1168 . 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 29901 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   CCcc 9279   0cc0 9281    x. cmul 9286    - cmin 9594   *ccj 12584   Imcim 12586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-2 10379  df-cj 12587  df-re 12588  df-im 12589
This theorem is referenced by: (None)
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