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Theorem cevath 43119
Description: Ceva's theorem. Let 𝐴𝐵𝐶 be a triangle and let points 𝐹, 𝐷 and 𝐸 lie on sides 𝐴𝐵, 𝐵𝐶, 𝐶𝐴 correspondingly. Suppose that cevians 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹 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 43118 three times and then using cevathlem1 43117 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 43114. This is Metamath 100 proof #61. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Hypotheses
Ref Expression
cevath.sigar 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
cevath.a (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))
cevath.b (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
cevath.c (𝜑𝑂 ∈ ℂ)
cevath.d (𝜑 → (((𝐴𝑂)𝐺(𝐷𝑂)) = 0 ∧ ((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0))
cevath.e (𝜑 → (((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0))
cevath.f (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
Assertion
Ref Expression
cevath (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
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:   𝜑(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem cevath
StepHypRef Expression
1 cevath.sigar . . . 4 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
2 cevath.a . . . . . . 7 (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))
32simp2d 1139 . . . . . 6 (𝜑𝐵 ∈ ℂ)
4 cevath.c . . . . . 6 (𝜑𝑂 ∈ ℂ)
53, 4subcld 10991 . . . . 5 (𝜑 → (𝐵𝑂) ∈ ℂ)
62simp3d 1140 . . . . . 6 (𝜑𝐶 ∈ ℂ)
76, 4subcld 10991 . . . . 5 (𝜑 → (𝐶𝑂) ∈ ℂ)
85, 7jca 514 . . . 4 (𝜑 → ((𝐵𝑂) ∈ ℂ ∧ (𝐶𝑂) ∈ ℂ))
91, 8sigarimcd 43112 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ)
102simp1d 1138 . . . 4 (𝜑𝐴 ∈ ℂ)
11 cevath.b . . . . 5 (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
1211simp1d 1138 . . . 4 (𝜑𝐹 ∈ ℂ)
1310, 12subcld 10991 . . 3 (𝜑 → (𝐴𝐹) ∈ ℂ)
1410, 4subcld 10991 . . . . 5 (𝜑 → (𝐴𝑂) ∈ ℂ)
157, 14jca 514 . . . 4 (𝜑 → ((𝐶𝑂) ∈ ℂ ∧ (𝐴𝑂) ∈ ℂ))
161, 15sigarimcd 43112 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ)
179, 13, 163jca 1124 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ ∧ (𝐴𝐹) ∈ ℂ ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ))
1812, 3subcld 10991 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
1914, 5jca 514 . . . 4 (𝜑 → ((𝐴𝑂) ∈ ℂ ∧ (𝐵𝑂) ∈ ℂ))
201, 19sigarimcd 43112 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ)
2111simp3d 1140 . . . 4 (𝜑𝐸 ∈ ℂ)
226, 21subcld 10991 . . 3 (𝜑 → (𝐶𝐸) ∈ ℂ)
2318, 20, 223jca 1124 . 2 (𝜑 → ((𝐹𝐵) ∈ ℂ ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ ∧ (𝐶𝐸) ∈ ℂ))
2421, 10subcld 10991 . . 3 (𝜑 → (𝐸𝐴) ∈ ℂ)
2511simp2d 1139 . . . 4 (𝜑𝐷 ∈ ℂ)
263, 25subcld 10991 . . 3 (𝜑 → (𝐵𝐷) ∈ ℂ)
2725, 6subcld 10991 . . 3 (𝜑 → (𝐷𝐶) ∈ ℂ)
2824, 26, 273jca 1124 . 2 (𝜑 → ((𝐸𝐴) ∈ ℂ ∧ (𝐵𝐷) ∈ ℂ ∧ (𝐷𝐶) ∈ ℂ))
29 cevath.f . . . 4 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
3029simp2d 1139 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0)
3129simp1d 1138 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0)
3229simp3d 1140 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0)
3330, 31, 323jca 1124 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
346, 10, 33jca 1124 . . . 4 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
3521, 12, 253jca 1124 . . . 4 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ))
36 cevath.d . . . . . 6 (𝜑 → (((𝐴𝑂)𝐺(𝐷𝑂)) = 0 ∧ ((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0))
3736simp3d 1140 . . . . 5 (𝜑 → ((𝐶𝑂)𝐺(𝐹𝑂)) = 0)
3836simp1d 1138 . . . . 5 (𝜑 → ((𝐴𝑂)𝐺(𝐷𝑂)) = 0)
3936simp2d 1139 . . . . 5 (𝜑 → ((𝐵𝑂)𝐺(𝐸𝑂)) = 0)
4037, 38, 393jca 1124 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0 ∧ ((𝐵𝑂)𝐺(𝐸𝑂)) = 0))
41 cevath.e . . . . . 6 (𝜑 → (((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0))
4241simp3d 1140 . . . . 5 (𝜑 → ((𝐶𝐸)𝐺(𝐴𝐸)) = 0)
4341simp1d 1138 . . . . 5 (𝜑 → ((𝐴𝐹)𝐺(𝐵𝐹)) = 0)
4441simp2d 1139 . . . . 5 (𝜑 → ((𝐵𝐷)𝐺(𝐶𝐷)) = 0)
4542, 43, 443jca 1124 . . . 4 (𝜑 → (((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0))
4632, 31, 303jca 1124 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0))
471, 34, 35, 4, 40, 45, 46cevathlem2 43118 . . 3 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)))
483, 6, 103jca 1124 . . . 4 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ))
4925, 21, 123jca 1124 . . . 4 (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
5039, 37, 383jca 1124 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0))
5144, 42, 433jca 1124 . . . 4 (𝜑 → (((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0))
5230, 32, 313jca 1124 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0))
531, 48, 49, 4, 50, 51, 52cevathlem2 43118 . . 3 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)))
541, 2, 11, 4, 36, 41, 29cevathlem2 43118 . . 3 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))
5547, 53, 543jca 1124 . 2 (𝜑 → ((((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)) ∧ (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)) ∧ (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶))))
5617, 23, 28, 33, 55cevathlem1 43117 1 (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083   = wceq 1533  wcel 2110  wne 3016  cfv 6350  (class class class)co 7150  cmpo 7152  cc 10529  0cc0 10531   · cmul 10536  cmin 10864  ccj 14449  cim 14451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-po 5469  df-so 5470  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-2 11694  df-cj 14452  df-re 14453  df-im 14454
This theorem is referenced by: (None)
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