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Theorem cevath 41989
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 41988 three times and then using cevathlem1 41987 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 41984. 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 1134 . . . . . 6 (𝜑𝐵 ∈ ℂ)
4 cevath.c . . . . . 6 (𝜑𝑂 ∈ ℂ)
53, 4subcld 10734 . . . . 5 (𝜑 → (𝐵𝑂) ∈ ℂ)
62simp3d 1135 . . . . . 6 (𝜑𝐶 ∈ ℂ)
76, 4subcld 10734 . . . . 5 (𝜑 → (𝐶𝑂) ∈ ℂ)
85, 7jca 507 . . . 4 (𝜑 → ((𝐵𝑂) ∈ ℂ ∧ (𝐶𝑂) ∈ ℂ))
91, 8sigarimcd 41982 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ)
102simp1d 1133 . . . 4 (𝜑𝐴 ∈ ℂ)
11 cevath.b . . . . 5 (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
1211simp1d 1133 . . . 4 (𝜑𝐹 ∈ ℂ)
1310, 12subcld 10734 . . 3 (𝜑 → (𝐴𝐹) ∈ ℂ)
1410, 4subcld 10734 . . . . 5 (𝜑 → (𝐴𝑂) ∈ ℂ)
157, 14jca 507 . . . 4 (𝜑 → ((𝐶𝑂) ∈ ℂ ∧ (𝐴𝑂) ∈ ℂ))
161, 15sigarimcd 41982 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ)
179, 13, 163jca 1119 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ ∧ (𝐴𝐹) ∈ ℂ ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ))
1812, 3subcld 10734 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
1914, 5jca 507 . . . 4 (𝜑 → ((𝐴𝑂) ∈ ℂ ∧ (𝐵𝑂) ∈ ℂ))
201, 19sigarimcd 41982 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ)
2111simp3d 1135 . . . 4 (𝜑𝐸 ∈ ℂ)
226, 21subcld 10734 . . 3 (𝜑 → (𝐶𝐸) ∈ ℂ)
2318, 20, 223jca 1119 . 2 (𝜑 → ((𝐹𝐵) ∈ ℂ ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ ∧ (𝐶𝐸) ∈ ℂ))
2421, 10subcld 10734 . . 3 (𝜑 → (𝐸𝐴) ∈ ℂ)
2511simp2d 1134 . . . 4 (𝜑𝐷 ∈ ℂ)
263, 25subcld 10734 . . 3 (𝜑 → (𝐵𝐷) ∈ ℂ)
2725, 6subcld 10734 . . 3 (𝜑 → (𝐷𝐶) ∈ ℂ)
2824, 26, 273jca 1119 . 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 1134 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0)
3129simp1d 1133 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0)
3229simp3d 1135 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0)
3330, 31, 323jca 1119 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
346, 10, 33jca 1119 . . . 4 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
3521, 12, 253jca 1119 . . . 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 1135 . . . . 5 (𝜑 → ((𝐶𝑂)𝐺(𝐹𝑂)) = 0)
3836simp1d 1133 . . . . 5 (𝜑 → ((𝐴𝑂)𝐺(𝐷𝑂)) = 0)
3936simp2d 1134 . . . . 5 (𝜑 → ((𝐵𝑂)𝐺(𝐸𝑂)) = 0)
4037, 38, 393jca 1119 . . . 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 1135 . . . . 5 (𝜑 → ((𝐶𝐸)𝐺(𝐴𝐸)) = 0)
4341simp1d 1133 . . . . 5 (𝜑 → ((𝐴𝐹)𝐺(𝐵𝐹)) = 0)
4441simp2d 1134 . . . . 5 (𝜑 → ((𝐵𝐷)𝐺(𝐶𝐷)) = 0)
4542, 43, 443jca 1119 . . . 4 (𝜑 → (((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0))
4632, 31, 303jca 1119 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0))
471, 34, 35, 4, 40, 45, 46cevathlem2 41988 . . 3 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)))
483, 6, 103jca 1119 . . . 4 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ))
4925, 21, 123jca 1119 . . . 4 (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
5039, 37, 383jca 1119 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0))
5144, 42, 433jca 1119 . . . 4 (𝜑 → (((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0))
5230, 32, 313jca 1119 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0))
531, 48, 49, 4, 50, 51, 52cevathlem2 41988 . . 3 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)))
541, 2, 11, 4, 36, 41, 29cevathlem2 41988 . . 3 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))
5547, 53, 543jca 1119 . 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 41987 1 (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1601  wcel 2107  wne 2969  cfv 6135  (class class class)co 6922  cmpt2 6924  cc 10270  0cc0 10272   · cmul 10277  cmin 10606  ccj 14243  cim 14245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-po 5274  df-so 5275  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-2 11438  df-cj 14246  df-re 14247  df-im 14248
This theorem is referenced by: (None)
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