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Theorem chordthm 20187
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 20186 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 2562 . . . 4  |-  ( ph  ->  P  =/=  D )
92, 3, 4, 5, 6, 8angpieqvd 20181 . . 3  |-  ( ph  ->  ( ( ( C  -  P ) F ( D  -  P
) )  =  pi  <->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) ) )
101, 9mpbid 201 . 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 2562 . . . . . 6  |-  ( ph  ->  P  =/=  B )
172, 12, 4, 13, 14, 16angpieqvd 20181 . . . . 5  |-  ( ph  ->  ( ( ( A  -  P ) F ( B  -  P
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )
1811, 17mpbid 201 . . . 4  |-  ( ph  ->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )
1918adantr 451 . . 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 706 . . . . . . 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 706 . . . . . . 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 2350 . . . . . 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 5915 . . . . 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 5915 . . . 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 706 . . . . 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 706 . . . . 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 706 . . . . 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 10782 . . . . . 6  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
32 simprl 732 . . . . . 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 3212 . . . . 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 733 . . . . 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 20186 . . . 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 706 . . . . 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 706 . . . . 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 736 . . . . . 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 3212 . . . . 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 737 . . . . 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 706 . . . . . 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 2350 . . . . 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 20186 . . . 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 2358 . . 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 2705 . 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 2705 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 358    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    \ cdif 3183   {csn 3674   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   CCcc 8780   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082    / cdiv 9468   2c2 9840   (,)cioo 10703   [,]cicc 10706   ^cexp 11151   Imcim 11630   abscabs 11766   picpi 12395   logclog 19965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396  df-sin 12398  df-cos 12399  df-pi 12401  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-limc 19269  df-dv 19270  df-log 19967
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