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Theorem chordthm 20666
Description: The intersecting chords theorem. If points A, B, C, and D lie on a circle (with center Q, say), and the point P is on the interior of the segments AB and CD, then the two products of lengths PA  x. PB and PC  x. PD are equal. The Euclidean plane is identified with the complex plane, and the fact that P is on AB and on CD is expressed by the hypothesis that the angles APB and CPD are equal to  pi. The result is proven by using chordthmlem5 20665 twice to show that PA  x. PB and PC  x. PD both equal BQ 2  - PQ 2 . This is similar to the proof of the theorem given in Euclid's Elements, where it is Proposition III.35. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
chordthm.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
chordthm.A  |-  ( ph  ->  A  e.  CC )
chordthm.B  |-  ( ph  ->  B  e.  CC )
chordthm.C  |-  ( ph  ->  C  e.  CC )
chordthm.D  |-  ( ph  ->  D  e.  CC )
chordthm.P  |-  ( ph  ->  P  e.  CC )
chordthm.AneP  |-  ( ph  ->  A  =/=  P )
chordthm.BneP  |-  ( ph  ->  B  =/=  P )
chordthm.CneP  |-  ( ph  ->  C  =/=  P )
chordthm.DneP  |-  ( ph  ->  D  =/=  P )
chordthm.APB  |-  ( ph  ->  ( ( A  -  P ) F ( B  -  P ) )  =  pi )
chordthm.CPD  |-  ( ph  ->  ( ( C  -  P ) F ( D  -  P ) )  =  pi )
chordthm.Q  |-  ( ph  ->  Q  e.  CC )
chordthm.ABcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
chordthm.ACcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( C  -  Q
) ) )
chordthm.ADcirc  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
Assertion
Ref Expression
chordthm  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, D, y   
x, P, y
Allowed substitution hints:    ph( x, y)    Q( x, y)    F( x, y)

Proof of Theorem chordthm
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chordthm.CPD . . 3  |-  ( ph  ->  ( ( C  -  P ) F ( D  -  P ) )  =  pi )
2 chordthm.angdef . . . 4  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
3 chordthm.C . . . 4  |-  ( ph  ->  C  e.  CC )
4 chordthm.P . . . 4  |-  ( ph  ->  P  e.  CC )
5 chordthm.D . . . 4  |-  ( ph  ->  D  e.  CC )
6 chordthm.CneP . . . 4  |-  ( ph  ->  C  =/=  P )
7 chordthm.DneP . . . . 5  |-  ( ph  ->  D  =/=  P )
87necomd 2681 . . . 4  |-  ( ph  ->  P  =/=  D )
92, 3, 4, 5, 6, 8angpieqvd 20660 . . 3  |-  ( ph  ->  ( ( ( C  -  P ) F ( D  -  P
) )  =  pi  <->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) ) )
101, 9mpbid 202 . 2  |-  ( ph  ->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) )
11 chordthm.APB . . . . 5  |-  ( ph  ->  ( ( A  -  P ) F ( B  -  P ) )  =  pi )
12 chordthm.A . . . . . 6  |-  ( ph  ->  A  e.  CC )
13 chordthm.B . . . . . 6  |-  ( ph  ->  B  e.  CC )
14 chordthm.AneP . . . . . 6  |-  ( ph  ->  A  =/=  P )
15 chordthm.BneP . . . . . . 7  |-  ( ph  ->  B  =/=  P )
1615necomd 2681 . . . . . 6  |-  ( ph  ->  P  =/=  B )
172, 12, 4, 13, 14, 16angpieqvd 20660 . . . . 5  |-  ( ph  ->  ( ( ( A  -  P ) F ( B  -  P
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )
1811, 17mpbid 202 . . . 4  |-  ( ph  ->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )
1918adantr 452 . . 3  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  ->  E. w  e.  (
0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )
20 chordthm.ABcirc . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
2120ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( B  -  Q
) ) )
22 chordthm.ADcirc . . . . . . . 8  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
2322ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
2421, 23eqtr3d 2469 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( abs `  ( B  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
2524oveq1d 6087 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( ( abs `  ( B  -  Q
) ) ^ 2 )  =  ( ( abs `  ( D  -  Q ) ) ^ 2 ) )
2625oveq1d 6087 . . . 4  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( ( ( abs `  ( B  -  Q ) ) ^ 2 )  -  ( ( abs `  ( P  -  Q )
) ^ 2 ) )  =  ( ( ( abs `  ( D  -  Q )
) ^ 2 )  -  ( ( abs `  ( P  -  Q
) ) ^ 2 ) ) )
2712ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  A  e.  CC )
2813ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  B  e.  CC )
29 chordthm.Q . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3029ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  Q  e.  CC )
31 ioossicc 10985 . . . . . 6  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
32 simprl 733 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  w  e.  ( 0 (,) 1 ) )
3331, 32sseldi 3338 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  w  e.  ( 0 [,] 1 ) )
34 simprr 734 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  P  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  B ) ) )
3527, 28, 30, 33, 34, 21chordthmlem5 20665 . . . 4  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( ( abs `  ( P  -  A
) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( ( abs `  ( B  -  Q ) ) ^ 2 )  -  ( ( abs `  ( P  -  Q )
) ^ 2 ) ) )
363ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  C  e.  CC )
375ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  D  e.  CC )
38 simplrl 737 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  v  e.  ( 0 (,) 1 ) )
3931, 38sseldi 3338 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  v  e.  ( 0 [,] 1 ) )
40 simplrr 738 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  P  =  ( ( v  x.  C
)  +  ( ( 1  -  v )  x.  D ) ) )
41 chordthm.ACcirc . . . . . . 7  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( C  -  Q
) ) )
4241ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( abs `  ( A  -  Q )
)  =  ( abs `  ( C  -  Q
) ) )
4342, 23eqtr3d 2469 . . . . 5  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( abs `  ( C  -  Q )
)  =  ( abs `  ( D  -  Q
) ) )
4436, 37, 30, 39, 40, 43chordthmlem5 20665 . . . 4  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( ( abs `  ( P  -  C
) )  x.  ( abs `  ( P  -  D ) ) )  =  ( ( ( abs `  ( D  -  Q ) ) ^ 2 )  -  ( ( abs `  ( P  -  Q )
) ^ 2 ) ) )
4526, 35, 443eqtr4d 2477 . . 3  |-  ( ( ( ph  /\  (
v  e.  ( 0 (,) 1 )  /\  P  =  ( (
v  x.  C )  +  ( ( 1  -  v )  x.  D ) ) ) )  /\  ( w  e.  ( 0 (,) 1 )  /\  P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )  ->  ( ( abs `  ( P  -  A
) )  x.  ( abs `  ( P  -  B ) ) )  =  ( ( abs `  ( P  -  C
) )  x.  ( abs `  ( P  -  D ) ) ) )
4619, 45rexlimddv 2826 . 2  |-  ( (
ph  /\  ( v  e.  ( 0 (,) 1
)  /\  P  =  ( ( v  x.  C )  +  ( ( 1  -  v
)  x.  D ) ) ) )  -> 
( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
4710, 46rexlimddv 2826 1  |-  ( ph  ->  ( ( abs `  ( P  -  A )
)  x.  ( abs `  ( P  -  B
) ) )  =  ( ( abs `  ( P  -  C )
)  x.  ( abs `  ( P  -  D
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309   {csn 3806   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   CCcc 8977   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    - cmin 9280    / cdiv 9666   2c2 10038   (,)cioo 10905   [,]cicc 10908   ^cexp 11370   Imcim 11891   abscabs 12027   picpi 12657   logclog 20440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442
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