MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chordthm Unicode version

Theorem chordthm 20097
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 20096 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
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 2504 . . . 4  |-  ( ph  ->  P  =/=  D )
92, 3, 4, 5, 6, 8angpieqvd 20091 . . 3  |-  ( ph  ->  ( ( ( C  -  P ) F ( D  -  P
) )  =  pi  <->  E. v  e.  ( 0 (,) 1 ) P  =  ( ( v  x.  C )  +  ( ( 1  -  v )  x.  D
) ) ) )
101, 9mpbid 203 . 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 2504 . . . . . 6  |-  ( ph  ->  P  =/=  B )
172, 12, 4, 13, 14, 16angpieqvd 20091 . . . . 5  |-  ( ph  ->  ( ( ( A  -  P ) F ( B  -  P
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B
) ) ) )
1811, 17mpbid 203 . . . 4  |-  ( ph  ->  E. w  e.  ( 0 (,) 1 ) P  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  B ) ) )
1918adantr 453 . . 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 709 . . . . . . 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 709 . . . . . . 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 2292 . . . . . 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 5807 . . . . 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 5807 . . . 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 709 . . . . 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 709 . . . . 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 709 . . . . 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 10702 . . . . . 6  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
32 simprl 735 . . . . . 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 3153 . . . . 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 736 . . . . 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 20096 . . . 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 709 . . . . 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 709 . . . . 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 739 . . . . . 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 3153 . . . . 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 740 . . . . 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 709 . . . . . 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 2292 . . . . 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 20096 . . . 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 2300 . . 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 2646 . 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 2646 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519    \ cdif 3124   {csn 3614   ` cfv 4673  (class class class)co 5792    e. cmpt2 5794   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005    / cdiv 9391   2c2 9763   (,)cioo 10623   [,]cicc 10626   ^cexp 11071   Imcim 11549   abscabs 11685   picpi 12311   logclog 19875
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-shft 11528  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-limsup 11911  df-clim 11928  df-rlim 11929  df-sum 12125  df-ef 12312  df-sin 12314  df-cos 12315  df-pi 12317  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-limc 19179  df-dv 19180  df-log 19877
  Copyright terms: Public domain W3C validator