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Theorem cevath 38350
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 38349 three times and then using cevathlem1 38348 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 38345. 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 1018 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 cevath.c . . . . . 6  |-  ( ph  ->  O  e.  CC )
53, 4subcld 9994 . . . . 5  |-  ( ph  ->  ( B  -  O
)  e.  CC )
62simp3d 1019 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76, 4subcld 9994 . . . . 5  |-  ( ph  ->  ( C  -  O
)  e.  CC )
85, 7jca 534 . . . 4  |-  ( ph  ->  ( ( B  -  O )  e.  CC  /\  ( C  -  O
)  e.  CC ) )
91, 8sigarimcd 38343 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  e.  CC )
102simp1d 1017 . . . 4  |-  ( ph  ->  A  e.  CC )
11 cevath.b . . . . 5  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
1211simp1d 1017 . . . 4  |-  ( ph  ->  F  e.  CC )
1310, 12subcld 9994 . . 3  |-  ( ph  ->  ( A  -  F
)  e.  CC )
1410, 4subcld 9994 . . . . 5  |-  ( ph  ->  ( A  -  O
)  e.  CC )
157, 14jca 534 . . . 4  |-  ( ph  ->  ( ( C  -  O )  e.  CC  /\  ( A  -  O
)  e.  CC ) )
161, 15sigarimcd 38343 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  e.  CC )
179, 13, 163jca 1185 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  e.  CC  /\  ( A  -  F
)  e.  CC  /\  ( ( C  -  O ) G ( A  -  O ) )  e.  CC ) )
1812, 3subcld 9994 . . 3  |-  ( ph  ->  ( F  -  B
)  e.  CC )
1914, 5jca 534 . . . 4  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
201, 19sigarimcd 38343 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
2111simp3d 1019 . . . 4  |-  ( ph  ->  E  e.  CC )
226, 21subcld 9994 . . 3  |-  ( ph  ->  ( C  -  E
)  e.  CC )
2318, 20, 223jca 1185 . 2  |-  ( ph  ->  ( ( F  -  B )  e.  CC  /\  ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  E
)  e.  CC ) )
2421, 10subcld 9994 . . 3  |-  ( ph  ->  ( E  -  A
)  e.  CC )
2511simp2d 1018 . . . 4  |-  ( ph  ->  D  e.  CC )
263, 25subcld 9994 . . 3  |-  ( ph  ->  ( B  -  D
)  e.  CC )
2725, 6subcld 9994 . . 3  |-  ( ph  ->  ( D  -  C
)  e.  CC )
2824, 26, 273jca 1185 . 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 1018 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 )
3129simp1d 1017 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 )
3229simp3d 1019 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 )
3330, 31, 323jca 1185 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
346, 10, 33jca 1185 . . . 4  |-  ( ph  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
3521, 12, 253jca 1185 . . . 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 1019 . . . . 5  |-  ( ph  ->  ( ( C  -  O ) G ( F  -  O ) )  =  0 )
3836simp1d 1017 . . . . 5  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
3936simp2d 1018 . . . . 5  |-  ( ph  ->  ( ( B  -  O ) G ( E  -  O ) )  =  0 )
4037, 38, 393jca 1185 . . . 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 1019 . . . . 5  |-  ( ph  ->  ( ( C  -  E ) G ( A  -  E ) )  =  0 )
4341simp1d 1017 . . . . 5  |-  ( ph  ->  ( ( A  -  F ) G ( B  -  F ) )  =  0 )
4441simp2d 1018 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
4542, 43, 443jca 1185 . . . 4  |-  ( ph  ->  ( ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0 ) )
4632, 31, 303jca 1185 . . . 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 38349 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  x.  ( A  -  F )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( F  -  B ) ) )
483, 6, 103jca 1185 . . . 4  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
4925, 21, 123jca 1185 . . . 4  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
5039, 37, 383jca 1185 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0 ) )
5144, 42, 433jca 1185 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0 ) )
5230, 32, 313jca 1185 . . . 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 38349 . . 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 38349 . . 3  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
5547, 53, 543jca 1185 . 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 38348 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 982    = wceq 1437    e. wcel 1872    =/= wne 2614   ` cfv 5601  (class class class)co 6306    |-> cmpt2 6308   CCcc 9545   0cc0 9547    x. cmul 9552    - cmin 9868   *ccj 13160   Imcim 13162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-2 10676  df-cj 13163  df-re 13164  df-im 13165
This theorem is referenced by: (None)
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