Users' Mathboxes Mathbox for Saveliy Skresanov < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cevath Structured version   Unicode version

Theorem cevath 27835
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 27834 three times and then using cevathlem1 27833 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 27830. (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 970 . . . . . 6  |-  ( ph  ->  B  e.  CC )
4 cevath.c . . . . . 6  |-  ( ph  ->  O  e.  CC )
53, 4subcld 9411 . . . . 5  |-  ( ph  ->  ( B  -  O
)  e.  CC )
62simp3d 971 . . . . . 6  |-  ( ph  ->  C  e.  CC )
76, 4subcld 9411 . . . . 5  |-  ( ph  ->  ( C  -  O
)  e.  CC )
85, 7jca 519 . . . 4  |-  ( ph  ->  ( ( B  -  O )  e.  CC  /\  ( C  -  O
)  e.  CC ) )
91, 8sigarimcd 27828 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  e.  CC )
102simp1d 969 . . . 4  |-  ( ph  ->  A  e.  CC )
11 cevath.b . . . . 5  |-  ( ph  ->  ( F  e.  CC  /\  D  e.  CC  /\  E  e.  CC )
)
1211simp1d 969 . . . 4  |-  ( ph  ->  F  e.  CC )
1310, 12subcld 9411 . . 3  |-  ( ph  ->  ( A  -  F
)  e.  CC )
1410, 4subcld 9411 . . . . 5  |-  ( ph  ->  ( A  -  O
)  e.  CC )
157, 14jca 519 . . . 4  |-  ( ph  ->  ( ( C  -  O )  e.  CC  /\  ( A  -  O
)  e.  CC ) )
161, 15sigarimcd 27828 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  e.  CC )
179, 13, 163jca 1134 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  e.  CC  /\  ( A  -  F
)  e.  CC  /\  ( ( C  -  O ) G ( A  -  O ) )  e.  CC ) )
1812, 3subcld 9411 . . 3  |-  ( ph  ->  ( F  -  B
)  e.  CC )
1914, 5jca 519 . . . 4  |-  ( ph  ->  ( ( A  -  O )  e.  CC  /\  ( B  -  O
)  e.  CC ) )
201, 19sigarimcd 27828 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  e.  CC )
2111simp3d 971 . . . 4  |-  ( ph  ->  E  e.  CC )
226, 21subcld 9411 . . 3  |-  ( ph  ->  ( C  -  E
)  e.  CC )
2318, 20, 223jca 1134 . 2  |-  ( ph  ->  ( ( F  -  B )  e.  CC  /\  ( ( A  -  O ) G ( B  -  O ) )  e.  CC  /\  ( C  -  E
)  e.  CC ) )
2421, 10subcld 9411 . . 3  |-  ( ph  ->  ( E  -  A
)  e.  CC )
2511simp2d 970 . . . 4  |-  ( ph  ->  D  e.  CC )
263, 25subcld 9411 . . 3  |-  ( ph  ->  ( B  -  D
)  e.  CC )
2725, 6subcld 9411 . . 3  |-  ( ph  ->  ( D  -  C
)  e.  CC )
2824, 26, 273jca 1134 . 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 970 . . 3  |-  ( ph  ->  ( ( B  -  O ) G ( C  -  O ) )  =/=  0 )
3129simp1d 969 . . 3  |-  ( ph  ->  ( ( A  -  O ) G ( B  -  O ) )  =/=  0 )
3229simp3d 971 . . 3  |-  ( ph  ->  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 )
3330, 31, 323jca 1134 . 2  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  =/=  0  /\  ( ( A  -  O ) G ( B  -  O ) )  =/=  0  /\  ( ( C  -  O ) G ( A  -  O ) )  =/=  0 ) )
346, 10, 33jca 1134 . . . 4  |-  ( ph  ->  ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )
)
3521, 12, 253jca 1134 . . . 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 971 . . . . 5  |-  ( ph  ->  ( ( C  -  O ) G ( F  -  O ) )  =  0 )
3836simp1d 969 . . . . 5  |-  ( ph  ->  ( ( A  -  O ) G ( D  -  O ) )  =  0 )
3936simp2d 970 . . . . 5  |-  ( ph  ->  ( ( B  -  O ) G ( E  -  O ) )  =  0 )
4037, 38, 393jca 1134 . . . 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 971 . . . . 5  |-  ( ph  ->  ( ( C  -  E ) G ( A  -  E ) )  =  0 )
4341simp1d 969 . . . . 5  |-  ( ph  ->  ( ( A  -  F ) G ( B  -  F ) )  =  0 )
4441simp2d 970 . . . . 5  |-  ( ph  ->  ( ( B  -  D ) G ( C  -  D ) )  =  0 )
4542, 43, 443jca 1134 . . . 4  |-  ( ph  ->  ( ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0  /\  ( ( B  -  D ) G ( C  -  D
) )  =  0 ) )
4632, 31, 303jca 1134 . . . 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 27834 . . 3  |-  ( ph  ->  ( ( ( B  -  O ) G ( C  -  O
) )  x.  ( A  -  F )
)  =  ( ( ( C  -  O
) G ( A  -  O ) )  x.  ( F  -  B ) ) )
483, 6, 103jca 1134 . . . 4  |-  ( ph  ->  ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )
)
4925, 21, 123jca 1134 . . . 4  |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
)
5039, 37, 383jca 1134 . . . 4  |-  ( ph  ->  ( ( ( B  -  O ) G ( E  -  O
) )  =  0  /\  ( ( C  -  O ) G ( F  -  O
) )  =  0  /\  ( ( A  -  O ) G ( D  -  O
) )  =  0 ) )
5144, 42, 433jca 1134 . . . 4  |-  ( ph  ->  ( ( ( B  -  D ) G ( C  -  D
) )  =  0  /\  ( ( C  -  E ) G ( A  -  E
) )  =  0  /\  ( ( A  -  F ) G ( B  -  F
) )  =  0 ) )
5230, 32, 313jca 1134 . . . 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 27834 . . 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 27834 . . 3  |-  ( ph  ->  ( ( ( C  -  O ) G ( A  -  O
) )  x.  ( B  -  D )
)  =  ( ( ( A  -  O
) G ( B  -  O ) )  x.  ( D  -  C ) ) )
5547, 53, 543jca 1134 . 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 27833 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 936    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   0cc0 8990    x. cmul 8995    - cmin 9291   *ccj 11901   Imcim 11903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-cj 11904  df-re 11905  df-im 11906
  Copyright terms: Public domain W3C validator