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Theorem cevath 46884
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 46883 three times and then using cevathlem1 46882 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 46879. 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 1144 . . . . . 6 (𝜑𝐵 ∈ ℂ)
4 cevath.c . . . . . 6 (𝜑𝑂 ∈ ℂ)
53, 4subcld 11620 . . . . 5 (𝜑 → (𝐵𝑂) ∈ ℂ)
62simp3d 1145 . . . . . 6 (𝜑𝐶 ∈ ℂ)
76, 4subcld 11620 . . . . 5 (𝜑 → (𝐶𝑂) ∈ ℂ)
85, 7jca 511 . . . 4 (𝜑 → ((𝐵𝑂) ∈ ℂ ∧ (𝐶𝑂) ∈ ℂ))
91, 8sigarimcd 46877 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ)
102simp1d 1143 . . . 4 (𝜑𝐴 ∈ ℂ)
11 cevath.b . . . . 5 (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
1211simp1d 1143 . . . 4 (𝜑𝐹 ∈ ℂ)
1310, 12subcld 11620 . . 3 (𝜑 → (𝐴𝐹) ∈ ℂ)
1410, 4subcld 11620 . . . . 5 (𝜑 → (𝐴𝑂) ∈ ℂ)
157, 14jca 511 . . . 4 (𝜑 → ((𝐶𝑂) ∈ ℂ ∧ (𝐴𝑂) ∈ ℂ))
161, 15sigarimcd 46877 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ)
179, 13, 163jca 1129 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ ∧ (𝐴𝐹) ∈ ℂ ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ))
1812, 3subcld 11620 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
1914, 5jca 511 . . . 4 (𝜑 → ((𝐴𝑂) ∈ ℂ ∧ (𝐵𝑂) ∈ ℂ))
201, 19sigarimcd 46877 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ)
2111simp3d 1145 . . . 4 (𝜑𝐸 ∈ ℂ)
226, 21subcld 11620 . . 3 (𝜑 → (𝐶𝐸) ∈ ℂ)
2318, 20, 223jca 1129 . 2 (𝜑 → ((𝐹𝐵) ∈ ℂ ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ ∧ (𝐶𝐸) ∈ ℂ))
2421, 10subcld 11620 . . 3 (𝜑 → (𝐸𝐴) ∈ ℂ)
2511simp2d 1144 . . . 4 (𝜑𝐷 ∈ ℂ)
263, 25subcld 11620 . . 3 (𝜑 → (𝐵𝐷) ∈ ℂ)
2725, 6subcld 11620 . . 3 (𝜑 → (𝐷𝐶) ∈ ℂ)
2824, 26, 273jca 1129 . 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 1144 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0)
3129simp1d 1143 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0)
3229simp3d 1145 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0)
3330, 31, 323jca 1129 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
346, 10, 33jca 1129 . . . 4 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
3521, 12, 253jca 1129 . . . 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 1145 . . . . 5 (𝜑 → ((𝐶𝑂)𝐺(𝐹𝑂)) = 0)
3836simp1d 1143 . . . . 5 (𝜑 → ((𝐴𝑂)𝐺(𝐷𝑂)) = 0)
3936simp2d 1144 . . . . 5 (𝜑 → ((𝐵𝑂)𝐺(𝐸𝑂)) = 0)
4037, 38, 393jca 1129 . . . 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 1145 . . . . 5 (𝜑 → ((𝐶𝐸)𝐺(𝐴𝐸)) = 0)
4341simp1d 1143 . . . . 5 (𝜑 → ((𝐴𝐹)𝐺(𝐵𝐹)) = 0)
4441simp2d 1144 . . . . 5 (𝜑 → ((𝐵𝐷)𝐺(𝐶𝐷)) = 0)
4542, 43, 443jca 1129 . . . 4 (𝜑 → (((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0))
4632, 31, 303jca 1129 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0))
471, 34, 35, 4, 40, 45, 46cevathlem2 46883 . . 3 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)))
483, 6, 103jca 1129 . . . 4 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ))
4925, 21, 123jca 1129 . . . 4 (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
5039, 37, 383jca 1129 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0))
5144, 42, 433jca 1129 . . . 4 (𝜑 → (((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0))
5230, 32, 313jca 1129 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0))
531, 48, 49, 4, 50, 51, 52cevathlem2 46883 . . 3 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)))
541, 2, 11, 4, 36, 41, 29cevathlem2 46883 . . 3 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))
5547, 53, 543jca 1129 . 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 46882 1 (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cfv 6561  (class class class)co 7431  cmpo 7433  cc 11153  0cc0 11155   · cmul 11160  cmin 11492  ccj 15135  cim 15137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-cj 15138  df-re 15139  df-im 15140
This theorem is referenced by: (None)
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