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Theorem cevath 46825
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 46824 three times and then using cevathlem1 46823 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 46820. 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 1142 . . . . . 6 (𝜑𝐵 ∈ ℂ)
4 cevath.c . . . . . 6 (𝜑𝑂 ∈ ℂ)
53, 4subcld 11618 . . . . 5 (𝜑 → (𝐵𝑂) ∈ ℂ)
62simp3d 1143 . . . . . 6 (𝜑𝐶 ∈ ℂ)
76, 4subcld 11618 . . . . 5 (𝜑 → (𝐶𝑂) ∈ ℂ)
85, 7jca 511 . . . 4 (𝜑 → ((𝐵𝑂) ∈ ℂ ∧ (𝐶𝑂) ∈ ℂ))
91, 8sigarimcd 46818 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ)
102simp1d 1141 . . . 4 (𝜑𝐴 ∈ ℂ)
11 cevath.b . . . . 5 (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
1211simp1d 1141 . . . 4 (𝜑𝐹 ∈ ℂ)
1310, 12subcld 11618 . . 3 (𝜑 → (𝐴𝐹) ∈ ℂ)
1410, 4subcld 11618 . . . . 5 (𝜑 → (𝐴𝑂) ∈ ℂ)
157, 14jca 511 . . . 4 (𝜑 → ((𝐶𝑂) ∈ ℂ ∧ (𝐴𝑂) ∈ ℂ))
161, 15sigarimcd 46818 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ)
179, 13, 163jca 1127 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ ∧ (𝐴𝐹) ∈ ℂ ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ))
1812, 3subcld 11618 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
1914, 5jca 511 . . . 4 (𝜑 → ((𝐴𝑂) ∈ ℂ ∧ (𝐵𝑂) ∈ ℂ))
201, 19sigarimcd 46818 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ)
2111simp3d 1143 . . . 4 (𝜑𝐸 ∈ ℂ)
226, 21subcld 11618 . . 3 (𝜑 → (𝐶𝐸) ∈ ℂ)
2318, 20, 223jca 1127 . 2 (𝜑 → ((𝐹𝐵) ∈ ℂ ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ ∧ (𝐶𝐸) ∈ ℂ))
2421, 10subcld 11618 . . 3 (𝜑 → (𝐸𝐴) ∈ ℂ)
2511simp2d 1142 . . . 4 (𝜑𝐷 ∈ ℂ)
263, 25subcld 11618 . . 3 (𝜑 → (𝐵𝐷) ∈ ℂ)
2725, 6subcld 11618 . . 3 (𝜑 → (𝐷𝐶) ∈ ℂ)
2824, 26, 273jca 1127 . 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 1142 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0)
3129simp1d 1141 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0)
3229simp3d 1143 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0)
3330, 31, 323jca 1127 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
346, 10, 33jca 1127 . . . 4 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
3521, 12, 253jca 1127 . . . 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 1143 . . . . 5 (𝜑 → ((𝐶𝑂)𝐺(𝐹𝑂)) = 0)
3836simp1d 1141 . . . . 5 (𝜑 → ((𝐴𝑂)𝐺(𝐷𝑂)) = 0)
3936simp2d 1142 . . . . 5 (𝜑 → ((𝐵𝑂)𝐺(𝐸𝑂)) = 0)
4037, 38, 393jca 1127 . . . 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 1143 . . . . 5 (𝜑 → ((𝐶𝐸)𝐺(𝐴𝐸)) = 0)
4341simp1d 1141 . . . . 5 (𝜑 → ((𝐴𝐹)𝐺(𝐵𝐹)) = 0)
4441simp2d 1142 . . . . 5 (𝜑 → ((𝐵𝐷)𝐺(𝐶𝐷)) = 0)
4542, 43, 443jca 1127 . . . 4 (𝜑 → (((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0))
4632, 31, 303jca 1127 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0))
471, 34, 35, 4, 40, 45, 46cevathlem2 46824 . . 3 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)))
483, 6, 103jca 1127 . . . 4 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ))
4925, 21, 123jca 1127 . . . 4 (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
5039, 37, 383jca 1127 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0))
5144, 42, 433jca 1127 . . . 4 (𝜑 → (((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0))
5230, 32, 313jca 1127 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0))
531, 48, 49, 4, 50, 51, 52cevathlem2 46824 . . 3 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)))
541, 2, 11, 4, 36, 41, 29cevathlem2 46824 . . 3 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))
5547, 53, 543jca 1127 . 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 46823 1 (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  cmpo 7433  cc 11151  0cc0 11153   · cmul 11158  cmin 11490  ccj 15132  cim 15134
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-2 12327  df-cj 15135  df-re 15136  df-im 15137
This theorem is referenced by: (None)
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