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

Theorem cevath 41571
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 41570 three times and then using cevathlem1 41569 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 41566. 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 1137 . . . . . 6 (𝜑𝐵 ∈ ℂ)
4 cevath.c . . . . . 6 (𝜑𝑂 ∈ ℂ)
53, 4subcld 10592 . . . . 5 (𝜑 → (𝐵𝑂) ∈ ℂ)
62simp3d 1138 . . . . . 6 (𝜑𝐶 ∈ ℂ)
76, 4subcld 10592 . . . . 5 (𝜑 → (𝐶𝑂) ∈ ℂ)
85, 7jca 501 . . . 4 (𝜑 → ((𝐵𝑂) ∈ ℂ ∧ (𝐶𝑂) ∈ ℂ))
91, 8sigarimcd 41564 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ)
102simp1d 1136 . . . 4 (𝜑𝐴 ∈ ℂ)
11 cevath.b . . . . 5 (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))
1211simp1d 1136 . . . 4 (𝜑𝐹 ∈ ℂ)
1310, 12subcld 10592 . . 3 (𝜑 → (𝐴𝐹) ∈ ℂ)
1410, 4subcld 10592 . . . . 5 (𝜑 → (𝐴𝑂) ∈ ℂ)
157, 14jca 501 . . . 4 (𝜑 → ((𝐶𝑂) ∈ ℂ ∧ (𝐴𝑂) ∈ ℂ))
161, 15sigarimcd 41564 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ)
179, 13, 163jca 1122 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ∈ ℂ ∧ (𝐴𝐹) ∈ ℂ ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ∈ ℂ))
1812, 3subcld 10592 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
1914, 5jca 501 . . . 4 (𝜑 → ((𝐴𝑂) ∈ ℂ ∧ (𝐵𝑂) ∈ ℂ))
201, 19sigarimcd 41564 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ)
2111simp3d 1138 . . . 4 (𝜑𝐸 ∈ ℂ)
226, 21subcld 10592 . . 3 (𝜑 → (𝐶𝐸) ∈ ℂ)
2318, 20, 223jca 1122 . 2 (𝜑 → ((𝐹𝐵) ∈ ℂ ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ∈ ℂ ∧ (𝐶𝐸) ∈ ℂ))
2421, 10subcld 10592 . . 3 (𝜑 → (𝐸𝐴) ∈ ℂ)
2511simp2d 1137 . . . 4 (𝜑𝐷 ∈ ℂ)
263, 25subcld 10592 . . 3 (𝜑 → (𝐵𝐷) ∈ ℂ)
2725, 6subcld 10592 . . 3 (𝜑 → (𝐷𝐶) ∈ ℂ)
2824, 26, 273jca 1122 . 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 1137 . . 3 (𝜑 → ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0)
3129simp1d 1136 . . 3 (𝜑 → ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0)
3229simp3d 1138 . . 3 (𝜑 → ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0)
3330, 31, 323jca 1122 . 2 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0))
346, 10, 33jca 1122 . . . 4 (𝜑 → (𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
3521, 12, 253jca 1122 . . . 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 1138 . . . . 5 (𝜑 → ((𝐶𝑂)𝐺(𝐹𝑂)) = 0)
3836simp1d 1136 . . . . 5 (𝜑 → ((𝐴𝑂)𝐺(𝐷𝑂)) = 0)
3936simp2d 1137 . . . . 5 (𝜑 → ((𝐵𝑂)𝐺(𝐸𝑂)) = 0)
4037, 38, 393jca 1122 . . . 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 1138 . . . . 5 (𝜑 → ((𝐶𝐸)𝐺(𝐴𝐸)) = 0)
4341simp1d 1136 . . . . 5 (𝜑 → ((𝐴𝐹)𝐺(𝐵𝐹)) = 0)
4441simp2d 1137 . . . . 5 (𝜑 → ((𝐵𝐷)𝐺(𝐶𝐷)) = 0)
4542, 43, 443jca 1122 . . . 4 (𝜑 → (((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0 ∧ ((𝐵𝐷)𝐺(𝐶𝐷)) = 0))
4632, 31, 303jca 1122 . . . 4 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0 ∧ ((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0))
471, 34, 35, 4, 40, 45, 46cevathlem2 41570 . . 3 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐴𝐹)) = (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐹𝐵)))
483, 6, 103jca 1122 . . . 4 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ))
4925, 21, 123jca 1122 . . . 4 (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
5039, 37, 383jca 1122 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐸𝑂)) = 0 ∧ ((𝐶𝑂)𝐺(𝐹𝑂)) = 0 ∧ ((𝐴𝑂)𝐺(𝐷𝑂)) = 0))
5144, 42, 433jca 1122 . . . 4 (𝜑 → (((𝐵𝐷)𝐺(𝐶𝐷)) = 0 ∧ ((𝐶𝐸)𝐺(𝐴𝐸)) = 0 ∧ ((𝐴𝐹)𝐺(𝐵𝐹)) = 0))
5230, 32, 313jca 1122 . . . 4 (𝜑 → (((𝐵𝑂)𝐺(𝐶𝑂)) ≠ 0 ∧ ((𝐶𝑂)𝐺(𝐴𝑂)) ≠ 0 ∧ ((𝐴𝑂)𝐺(𝐵𝑂)) ≠ 0))
531, 48, 49, 4, 50, 51, 52cevathlem2 41570 . . 3 (𝜑 → (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐶𝐸)) = (((𝐵𝑂)𝐺(𝐶𝑂)) · (𝐸𝐴)))
541, 2, 11, 4, 36, 41, 29cevathlem2 41570 . . 3 (𝜑 → (((𝐶𝑂)𝐺(𝐴𝑂)) · (𝐵𝐷)) = (((𝐴𝑂)𝐺(𝐵𝑂)) · (𝐷𝐶)))
5547, 53, 543jca 1122 . 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 41569 1 (𝜑 → (((𝐴𝐹) · (𝐶𝐸)) · (𝐵𝐷)) = (((𝐹𝐵) · (𝐸𝐴)) · (𝐷𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wne 2943  cfv 6029  (class class class)co 6791  cmpt2 6793  cc 10134  0cc0 10136   · cmul 10141  cmin 10466  ccj 14037  cim 14039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-div 10885  df-2 11279  df-cj 14040  df-re 14041  df-im 14042
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator