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Theorem chordthm 20631
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 20630 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 2650 . . . 4  |-  ( ph  ->  P  =/=  D )
92, 3, 4, 5, 6, 8angpieqvd 20625 . . 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 2650 . . . . . 6  |-  ( ph  ->  P  =/=  B )
172, 12, 4, 13, 14, 16angpieqvd 20625 . . . . 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 2438 . . . . . 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 6055 . . . . 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 6055 . . . 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 10952 . . . . . 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 3306 . . . . 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 20630 . . . 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 3306 . . . . 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 2438 . . . . 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 20630 . . . 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 2446 . . 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 2794 . 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 2794 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633   2c2 10005   (,)cioo 10872   [,]cicc 10875   ^cexp 11337   Imcim 11858   abscabs 11994   picpi 12624   logclog 20405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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