:: Angle and Triangle in {E}uclidian Topological Space
:: by Akihiro Kubo and Yatsuka Nakamura
::
:: Received May 29, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

definition

let z be Complex;

func cpx2euc z -> Point of (TOP-REAL 2) equals :: EUCLID_3:def 1
|[(Re z),(Im z)]|;

correctness
coherence
|[(Re z),(Im z)]| is Point of (TOP-REAL 2);
;

end;

:: deftheorem defines cpx2euc EUCLID_3:def 1 :
for z being Complex holds cpx2euc z = |[(Re z),(Im z)]|;

definition

let p be Point of (TOP-REAL 2);

func euc2cpx p -> Element of COMPLEX equals :: EUCLID_3:def 2
(p `1) + ((p `2) * <i>);

correctness
coherence
(p `1) + ((p `2) * <i>) is Element of COMPLEX ;
by XCMPLX_0:def 2;

end;

:: deftheorem defines euc2cpx EUCLID_3:def 2 :
for p being Point of (TOP-REAL 2) holds euc2cpx p = (p `1) + ((p `2) * <i>);

theorem Th1: :: EUCLID_3:1

for z being Complex holds euc2cpx (cpx2euc z) = z

proof end;

theorem Th2: :: EUCLID_3:2

for p being Point of (TOP-REAL 2) holds cpx2euc (euc2cpx p) = p

proof end;

theorem :: EUCLID_3:3

for z1, z2 being Complex st cpx2euc z1 = cpx2euc z2 holds
z1 = z2

proof end;

theorem Th4: :: EUCLID_3:4

for p1, p2 being Point of (TOP-REAL 2) st euc2cpx p1 = euc2cpx p2 holds
p1 = p2

proof end;

theorem Th5: :: EUCLID_3:5

for x1, x2 being Real holds cpx2euc (x1 + (x2 * <i>)) = |[x1,x2]|

proof end;

theorem Th6: :: EUCLID_3:6

for z1, z2 being Complex holds |[(Re (z1 + z2)),(Im (z1 + z2))]| = |[((Re z1) + (Re z2)),((Im z1) + (Im z2))]|

proof end;

theorem Th7: :: EUCLID_3:7

for z1, z2 being Complex holds cpx2euc (z1 + z2) = (cpx2euc z1) + (cpx2euc z2)

proof end;

theorem Th8: :: EUCLID_3:8

for p1, p2 being Point of (TOP-REAL 2) holds ((p1 + p2) `1) + (((p1 + p2) `2) * <i>) = ((p1 `1) + (p2 `1)) + (((p1 `2) + (p2 `2)) * <i>)

proof end;

theorem Th9: :: EUCLID_3:9

for p1, p2 being Point of (TOP-REAL 2) holds euc2cpx (p1 + p2) = (euc2cpx p1) + (euc2cpx p2)

proof end;

theorem Th10: :: EUCLID_3:10

for z being Complex holds |[(Re (- z)),(Im (- z))]| = |[(- (Re z)),(- (Im z))]|

proof end;

theorem Th11: :: EUCLID_3:11

for z being Complex holds cpx2euc (- z) = - (cpx2euc z)

proof end;

theorem Th12: :: EUCLID_3:12

for p being Point of (TOP-REAL 2) holds ((- p) `1) + (((- p) `2) * <i>) = (- (p `1)) + ((- (p `2)) * <i>)

proof end;

theorem Th13: :: EUCLID_3:13

for p being Point of (TOP-REAL 2) holds euc2cpx (- p) = - (euc2cpx p)

proof end;

theorem :: EUCLID_3:14

for z1, z2 being Complex holds cpx2euc (z1 - z2) = (cpx2euc z1) - (cpx2euc z2)

proof end;

theorem Th15: :: EUCLID_3:15

for p1, p2 being Point of (TOP-REAL 2) holds euc2cpx (p1 - p2) = (euc2cpx p1) - (euc2cpx p2)

proof end;

theorem Th16: :: EUCLID_3:16

cpx2euc 0c = 0. (TOP-REAL 2) by COMPLEX1:4, EUCLID:54;

theorem Th17: :: EUCLID_3:17

euc2cpx (0. (TOP-REAL 2)) = 0c by Th1, Th16;

theorem :: EUCLID_3:18

for p being Point of (TOP-REAL 2) st euc2cpx p = 0c holds
p = 0. (TOP-REAL 2) by Th2, Th16;

theorem :: EUCLID_3:19

for z being Complex
for r being Real holds cpx2euc (r * z) = r * (cpx2euc z)

proof end;

theorem :: EUCLID_3:20

for r being Real
for p being Point of (TOP-REAL 2) holds euc2cpx (r * p) = r * (euc2cpx p)

proof end;

theorem Th21: :: EUCLID_3:21

for p being Point of (TOP-REAL 2) holds |.(euc2cpx p).| = sqrt (((p `1) ^2) + ((p `2) ^2))

proof end;

theorem :: EUCLID_3:22

for f being FinSequence of REAL st len f = 2 holds
|.f.| = sqrt (((f . 1) ^2) + ((f . 2) ^2))

proof end;

theorem :: EUCLID_3:23

for f being FinSequence of REAL
for p being Point of (TOP-REAL 2) st len f = 2 & p = f holds
|.p.| = |.f.| ;

theorem :: EUCLID_3:24

for z being Complex holds |.(cpx2euc z).| = sqrt (((Re z) ^2) + ((Im z) ^2))

proof end;

theorem Th25: :: EUCLID_3:25

for p being Point of (TOP-REAL 2) holds |.(euc2cpx p).| = |.p.|

proof end;

definition

let p be Point of (TOP-REAL 2);

func Arg p -> Real equals :: EUCLID_3:def 3
Arg (euc2cpx p);

correctness
coherence
Arg (euc2cpx p) is Real;
;

end;

:: deftheorem defines Arg EUCLID_3:def 3 :
for p being Point of (TOP-REAL 2) holds Arg p = Arg (euc2cpx p);

theorem :: EUCLID_3:26

for z being Element of COMPLEX
for p being Point of (TOP-REAL 2) st ( z = euc2cpx p or p = cpx2euc z ) holds
Arg z = Arg p

proof end;

theorem :: EUCLID_3:27

for x1, x2 being Real
for p being Point of (TOP-REAL 2) st x1 = |.p.| * (cos (Arg p)) & x2 = |.p.| * (sin (Arg p)) holds
p = |[x1,x2]|

proof end;

theorem :: EUCLID_3:28

Arg (0. (TOP-REAL 2)) = 0 by Th17, COMPTRIG:35;

theorem :: EUCLID_3:29

for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( Arg p < PI implies Arg (- p) = (Arg p) + PI ) & ( Arg p >= PI implies Arg (- p) = (Arg p) - PI ) )

proof end;

theorem :: EUCLID_3:30

for p being Point of (TOP-REAL 2) st Arg p = 0 holds
( p = |[|.p.|,0]| & p `2 = 0 )

proof end;

theorem Th31: :: EUCLID_3:31

for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( Arg p < PI iff Arg (- p) >= PI )

proof end;

theorem :: EUCLID_3:32

for p1, p2 being Point of (TOP-REAL 2) st ( p1 <> p2 or p1 - p2 <> 0. (TOP-REAL 2) ) holds
( Arg (p1 - p2) < PI iff Arg (p2 - p1) >= PI )

proof end;

theorem :: EUCLID_3:33

for p being Point of (TOP-REAL 2) holds
( Arg p in ].0,PI.[ iff p `2 > 0 )

proof end;

theorem :: EUCLID_3:34

for p1, p2 being Point of (TOP-REAL 2) st Arg p1 < PI & Arg p2 < PI holds
Arg (p1 + p2) < PI

proof end;

definition

let p1, p2, p3 be Point of (TOP-REAL 2);

func angle (p1,p2,p3) -> Real equals :: EUCLID_3:def 4
angle ((euc2cpx p1),(euc2cpx p2),(euc2cpx p3));

correctness
coherence
angle ((euc2cpx p1),(euc2cpx p2),(euc2cpx p3)) is Real;
;

end;

:: deftheorem defines angle EUCLID_3:def 4 :
for p1, p2, p3 being Point of (TOP-REAL 2) holds angle (p1,p2,p3) = angle ((euc2cpx p1),(euc2cpx p2),(euc2cpx p3));

theorem :: EUCLID_3:35

for p1, p2, p3 being Point of (TOP-REAL 2) holds angle (p1,p2,p3) = angle ((p1 - p2),(0. (TOP-REAL 2)),(p3 - p2))

proof end;

theorem :: EUCLID_3:36

for p1, p2, p3 being Point of (TOP-REAL 2) st angle (p1,p2,p3) = 0 holds
( Arg (p1 - p2) = Arg (p3 - p2) & angle (p3,p2,p1) = 0 )

proof end;

theorem :: EUCLID_3:37

for p1, p2, p3 being Point of (TOP-REAL 2) st angle (p1,p2,p3) <> 0 holds
angle (p3,p2,p1) = (2 * PI) - (angle (p1,p2,p3))

proof end;

theorem :: EUCLID_3:38

for p1, p2, p3 being Point of (TOP-REAL 2) st angle (p3,p2,p1) <> 0 holds
angle (p3,p2,p1) = (2 * PI) - (angle (p1,p2,p3))

proof end;

theorem Th39: :: EUCLID_3:39

for x, y being Element of COMPLEX holds Re (x .|. y) = ((Re x) * (Re y)) + ((Im x) * (Im y))

proof end;

theorem Th40: :: EUCLID_3:40

for x, y being Element of COMPLEX holds Im (x .|. y) = (- ((Re x) * (Im y))) + ((Im x) * (Re y))

proof end;

theorem Th41: :: EUCLID_3:41

for p, q being Point of (TOP-REAL 2) holds |(p,q)| = ((p `1) * (q `1)) + ((p `2) * (q `2))

proof end;

theorem Th42: :: EUCLID_3:42

for p1, p2 being Point of (TOP-REAL 2) holds |(p1,p2)| = Re ((euc2cpx p1) .|. (euc2cpx p2))

proof end;

theorem :: EUCLID_3:43

for p1, p2, p3 being Point of (TOP-REAL 2) st p1 <> 0. (TOP-REAL 2) & p2 <> 0. (TOP-REAL 2) holds
( |(p1,p2)| = 0 iff ( angle (p1,(0. (TOP-REAL 2)),p2) = PI / 2 or angle (p1,(0. (TOP-REAL 2)),p2) = (3 / 2) * PI ) )

proof end;

theorem :: EUCLID_3:44

for p1, p2 being Point of (TOP-REAL 2) st p1 <> 0. (TOP-REAL 2) & p2 <> 0. (TOP-REAL 2) holds
( ( ( not (- ((p1 `1) * (p2 `2))) + ((p1 `2) * (p2 `1)) = |.p1.| * |.p2.| & not (- ((p1 `1) * (p2 `2))) + ((p1 `2) * (p2 `1)) = - (|.p1.| * |.p2.|) ) or angle (p1,(0. (TOP-REAL 2)),p2) = PI / 2 or angle (p1,(0. (TOP-REAL 2)),p2) = (3 / 2) * PI ) & ( ( not angle (p1,(0. (TOP-REAL 2)),p2) = PI / 2 & not angle (p1,(0. (TOP-REAL 2)),p2) = (3 / 2) * PI ) or (- ((p1 `1) * (p2 `2))) + ((p1 `2) * (p2 `1)) = |.p1.| * |.p2.| or (- ((p1 `1) * (p2 `2))) + ((p1 `2) * (p2 `1)) = - (|.p1.| * |.p2.|) ) )

proof end;

theorem :: EUCLID_3:45

for p1, p2, p3 being Point of (TOP-REAL 2) st p1 <> p2 & p3 <> p2 holds
( |((p1 - p2),(p3 - p2))| = 0 iff ( angle (p1,p2,p3) = PI / 2 or angle (p1,p2,p3) = (3 / 2) * PI ) )

proof end;

:: Pythagorean Theorem

theorem :: EUCLID_3:46

for p1, p2, p3 being Point of (TOP-REAL 2) st p1 <> p2 & p3 <> p2 & ( angle (p1,p2,p3) = PI / 2 or angle (p1,p2,p3) = (3 / 2) * PI ) holds
(|.(p1 - p2).| ^2) + (|.(p3 - p2).| ^2) = |.(p1 - p3).| ^2

proof end;

theorem :: EUCLID_3:47

for p1, p2, p3 being Point of (TOP-REAL 2) st p2 <> p1 & p1 <> p3 & p3 <> p2 & angle (p2,p1,p3) < PI holds
((angle (p2,p1,p3)) + (angle (p1,p3,p2))) + (angle (p3,p2,p1)) = PI

proof end;

definition

let n be Element of NAT ;
let p1, p2, p3 be Point of (TOP-REAL n);

func Triangle (p1,p2,p3) -> Subset of (TOP-REAL n) equals :: EUCLID_3:def 5
((LSeg (p1,p2)) \/ (LSeg (p2,p3))) \/ (LSeg (p3,p1));

correctness
coherence
((LSeg (p1,p2)) \/ (LSeg (p2,p3))) \/ (LSeg (p3,p1)) is Subset of (TOP-REAL n);
;

end;

:: deftheorem defines Triangle EUCLID_3:def 5 :
for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds Triangle (p1,p2,p3) = ((LSeg (p1,p2)) \/ (LSeg (p2,p3))) \/ (LSeg (p3,p1));

definition

let n be Element of NAT ;
let p1, p2, p3 be Point of (TOP-REAL n);

func closed_inside_of_triangle (p1,p2,p3) -> Subset of (TOP-REAL n) equals :: EUCLID_3:def 6
{ p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( 0 <= a1 & 0 <= a2 & 0 <= a3 & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } ;

correctness
coherence
{ p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( 0 <= a1 & 0 <= a2 & 0 <= a3 & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } is Subset of (TOP-REAL n);

proof end;

end;

:: deftheorem defines closed_inside_of_triangle EUCLID_3:def 6 :
for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds closed_inside_of_triangle (p1,p2,p3) = { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( 0 <= a1 & 0 <= a2 & 0 <= a3 & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } ;

definition

let n be Element of NAT ;
let p1, p2, p3 be Point of (TOP-REAL n);

func inside_of_triangle (p1,p2,p3) -> Subset of (TOP-REAL n) equals :: EUCLID_3:def 7
(closed_inside_of_triangle (p1,p2,p3)) \ (Triangle (p1,p2,p3));

correctness
coherence
(closed_inside_of_triangle (p1,p2,p3)) \ (Triangle (p1,p2,p3)) is Subset of (TOP-REAL n);
;

end;

:: deftheorem defines inside_of_triangle EUCLID_3:def 7 :
for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds inside_of_triangle (p1,p2,p3) = (closed_inside_of_triangle (p1,p2,p3)) \ (Triangle (p1,p2,p3));

definition

let n be Element of NAT ;
let p1, p2, p3 be Point of (TOP-REAL n);

func outside_of_triangle (p1,p2,p3) -> Subset of (TOP-REAL n) equals :: EUCLID_3:def 8
{ p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( ( 0 > a1 or 0 > a2 or 0 > a3 ) & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } ;

correctness
coherence
{ p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( ( 0 > a1 or 0 > a2 or 0 > a3 ) & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } is Subset of (TOP-REAL n);

proof end;

end;

:: deftheorem defines outside_of_triangle EUCLID_3:def 8 :
for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds outside_of_triangle (p1,p2,p3) = { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( ( 0 > a1 or 0 > a2 or 0 > a3 ) & (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) } ;

definition

let n be Element of NAT ;
let p1, p2, p3 be Point of (TOP-REAL n);

func plane (p1,p2,p3) -> Subset of (TOP-REAL n) equals :: EUCLID_3:def 9
(outside_of_triangle (p1,p2,p3)) \/ (closed_inside_of_triangle (p1,p2,p3));

correctness
coherence
(outside_of_triangle (p1,p2,p3)) \/ (closed_inside_of_triangle (p1,p2,p3)) is Subset of (TOP-REAL n);
;

end;

:: deftheorem defines plane EUCLID_3:def 9 :
for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds plane (p1,p2,p3) = (outside_of_triangle (p1,p2,p3)) \/ (closed_inside_of_triangle (p1,p2,p3));

theorem Th48: :: EUCLID_3:48

for n being Element of NAT
for p1, p2, p3, p being Point of (TOP-REAL n) st p in plane (p1,p2,p3) holds
ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) )

proof end;

theorem :: EUCLID_3:49

for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds Triangle (p1,p2,p3) c= closed_inside_of_triangle (p1,p2,p3)

proof end;

definition

let n be Element of NAT ;
let q1, q2 be Point of (TOP-REAL n);

pred q1,q2 are_lindependent2 means :: EUCLID_3:def 10
for a1, a2 being Real st (a1 * q1) + (a2 * q2) = 0. (TOP-REAL n) holds
( a1 = 0 & a2 = 0 );

end;

:: deftheorem defines are_lindependent2 EUCLID_3:def 10 :
for n being Element of NAT
for q1, q2 being Point of (TOP-REAL n) holds
( q1,q2 are_lindependent2 iff for a1, a2 being Real st (a1 * q1) + (a2 * q2) = 0. (TOP-REAL n) holds
( a1 = 0 & a2 = 0 ) );

notation

let n be Element of NAT ;
let q1, q2 be Point of (TOP-REAL n);
antonym q1,q2 are_ldependent2 for q1,q2 are_lindependent2 ;

end;

theorem Th50: :: EUCLID_3:50

for n being Element of NAT
for q1, q2 being Point of (TOP-REAL n) st q1,q2 are_lindependent2 holds
( q1 <> q2 & q1 <> 0. (TOP-REAL n) & q2 <> 0. (TOP-REAL n) )

proof end;

theorem Th51: :: EUCLID_3:51

for n being Element of NAT
for p1, p2, p3, p0 being Point of (TOP-REAL n) st p2 - p1,p3 - p1 are_lindependent2 & p0 in plane (p1,p2,p3) holds
ex a1, a2, a3 being Real st
( p0 = ((a1 * p1) + (a2 * p2)) + (a3 * p3) & (a1 + a2) + a3 = 1 & ( for b1, b2, b3 being Real st p0 = ((b1 * p1) + (b2 * p2)) + (b3 * p3) & (b1 + b2) + b3 = 1 holds
( b1 = a1 & b2 = a2 & b3 = a3 ) ) )

proof end;

theorem Th52: :: EUCLID_3:52

for n being Element of NAT
for p1, p2, p3, p0 being Point of (TOP-REAL n) st ex a1, a2, a3 being Real st
( p0 = ((a1 * p1) + (a2 * p2)) + (a3 * p3) & (a1 + a2) + a3 = 1 ) holds
p0 in plane (p1,p2,p3)

proof end;

theorem :: EUCLID_3:53

for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds plane (p1,p2,p3) = { p where p is Point of (TOP-REAL n) : ex a1, a2, a3 being Real st
( (a1 + a2) + a3 = 1 & p = ((a1 * p1) + (a2 * p2)) + (a3 * p3) ) }

proof end;

theorem Th54: :: EUCLID_3:54

for p1, p2, p3 being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
plane (p1,p2,p3) = REAL 2

proof end;

definition

let n be Element of NAT ;
let p1, p2, p3, p be Point of (TOP-REAL n);
assume A1: ( p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) ) ;

func tricord1 (p1,p2,p3,p) -> Real means :Def11: :: EUCLID_3:def 11
ex a2, a3 being Real st
( (it + a2) + a3 = 1 & p = ((it * p1) + (a2 * p2)) + (a3 * p3) );

existence
ex b₁, a2, a3 being Real st
( (b₁ + a2) + a3 = 1 & p = ((b₁ * p1) + (a2 * p2)) + (a3 * p3) )

proof end;

uniqueness
for b₁, b₂ being Real st ex a2, a3 being Real st
( (b₁ + a2) + a3 = 1 & p = ((b₁ * p1) + (a2 * p2)) + (a3 * p3) ) & ex a2, a3 being Real st
( (b₂ + a2) + a3 = 1 & p = ((b₂ * p1) + (a2 * p2)) + (a3 * p3) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines tricord1 EUCLID_3:def 11 :
for n being Element of NAT
for p1, p2, p3, p being Point of (TOP-REAL n) st p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) holds
for b₆ being Real holds
( b₆ = tricord1 (p1,p2,p3,p) iff ex a2, a3 being Real st
( (b₆ + a2) + a3 = 1 & p = ((b₆ * p1) + (a2 * p2)) + (a3 * p3) ) );

definition

let n be Element of NAT ;
let p1, p2, p3, p be Point of (TOP-REAL n);
assume A1: ( p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) ) ;

func tricord2 (p1,p2,p3,p) -> Real means :Def12: :: EUCLID_3:def 12
ex a1, a3 being Real st
( (a1 + it) + a3 = 1 & p = ((a1 * p1) + (it * p2)) + (a3 * p3) );

existence
ex b₁, a1, a3 being Real st
( (a1 + b₁) + a3 = 1 & p = ((a1 * p1) + (b₁ * p2)) + (a3 * p3) )

proof end;

uniqueness
for b₁, b₂ being Real st ex a1, a3 being Real st
( (a1 + b₁) + a3 = 1 & p = ((a1 * p1) + (b₁ * p2)) + (a3 * p3) ) & ex a1, a3 being Real st
( (a1 + b₂) + a3 = 1 & p = ((a1 * p1) + (b₂ * p2)) + (a3 * p3) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines tricord2 EUCLID_3:def 12 :
for n being Element of NAT
for p1, p2, p3, p being Point of (TOP-REAL n) st p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) holds
for b₆ being Real holds
( b₆ = tricord2 (p1,p2,p3,p) iff ex a1, a3 being Real st
( (a1 + b₆) + a3 = 1 & p = ((a1 * p1) + (b₆ * p2)) + (a3 * p3) ) );

definition

let n be Element of NAT ;
let p1, p2, p3, p be Point of (TOP-REAL n);
assume A1: ( p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) ) ;

func tricord3 (p1,p2,p3,p) -> Real means :Def13: :: EUCLID_3:def 13
ex a1, a2 being Real st
( (a1 + a2) + it = 1 & p = ((a1 * p1) + (a2 * p2)) + (it * p3) );

existence
ex b₁, a1, a2 being Real st
( (a1 + a2) + b₁ = 1 & p = ((a1 * p1) + (a2 * p2)) + (b₁ * p3) )

proof end;

uniqueness
for b₁, b₂ being Real st ex a1, a2 being Real st
( (a1 + a2) + b₁ = 1 & p = ((a1 * p1) + (a2 * p2)) + (b₁ * p3) ) & ex a1, a2 being Real st
( (a1 + a2) + b₂ = 1 & p = ((a1 * p1) + (a2 * p2)) + (b₂ * p3) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines tricord3 EUCLID_3:def 13 :
for n being Element of NAT
for p1, p2, p3, p being Point of (TOP-REAL n) st p2 - p1,p3 - p1 are_lindependent2 & p in plane (p1,p2,p3) holds
for b₆ being Real holds
( b₆ = tricord3 (p1,p2,p3,p) iff ex a1, a2 being Real st
( (a1 + a2) + b₆ = 1 & p = ((a1 * p1) + (a2 * p2)) + (b₆ * p3) ) );

definition

let p1, p2, p3 be Point of (TOP-REAL 2);

func trcmap1 (p1,p2,p3) -> Function of (TOP-REAL 2),R^1 means :: EUCLID_3:def 14
for p being Point of (TOP-REAL 2) holds it . p = tricord1 (p1,p2,p3,p);

existence
ex b₁ being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b₁ . p = tricord1 (p1,p2,p3,p)

proof end;

uniqueness
for b₁, b₂ being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b₁ . p = tricord1 (p1,p2,p3,p) ) & ( for p being Point of (TOP-REAL 2) holds b₂ . p = tricord1 (p1,p2,p3,p) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines trcmap1 EUCLID_3:def 14 :
for p1, p2, p3 being Point of (TOP-REAL 2)
for b₄ being Function of (TOP-REAL 2),R^1 holds
( b₄ = trcmap1 (p1,p2,p3) iff for p being Point of (TOP-REAL 2) holds b₄ . p = tricord1 (p1,p2,p3,p) );

definition

let p1, p2, p3 be Point of (TOP-REAL 2);

func trcmap2 (p1,p2,p3) -> Function of (TOP-REAL 2),R^1 means :: EUCLID_3:def 15
for p being Point of (TOP-REAL 2) holds it . p = tricord2 (p1,p2,p3,p);

existence
ex b₁ being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b₁ . p = tricord2 (p1,p2,p3,p)

proof end;

uniqueness
for b₁, b₂ being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b₁ . p = tricord2 (p1,p2,p3,p) ) & ( for p being Point of (TOP-REAL 2) holds b₂ . p = tricord2 (p1,p2,p3,p) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines trcmap2 EUCLID_3:def 15 :
for p1, p2, p3 being Point of (TOP-REAL 2)
for b₄ being Function of (TOP-REAL 2),R^1 holds
( b₄ = trcmap2 (p1,p2,p3) iff for p being Point of (TOP-REAL 2) holds b₄ . p = tricord2 (p1,p2,p3,p) );

definition

let p1, p2, p3 be Point of (TOP-REAL 2);

func trcmap3 (p1,p2,p3) -> Function of (TOP-REAL 2),R^1 means :: EUCLID_3:def 16
for p being Point of (TOP-REAL 2) holds it . p = tricord3 (p1,p2,p3,p);

existence
ex b₁ being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b₁ . p = tricord3 (p1,p2,p3,p)

proof end;

uniqueness
for b₁, b₂ being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b₁ . p = tricord3 (p1,p2,p3,p) ) & ( for p being Point of (TOP-REAL 2) holds b₂ . p = tricord3 (p1,p2,p3,p) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem defines trcmap3 EUCLID_3:def 16 :
for p1, p2, p3 being Point of (TOP-REAL 2)
for b₄ being Function of (TOP-REAL 2),R^1 holds
( b₄ = trcmap3 (p1,p2,p3) iff for p being Point of (TOP-REAL 2) holds b₄ . p = tricord3 (p1,p2,p3,p) );

theorem :: EUCLID_3:55

for p1, p2, p3, p being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
( p in outside_of_triangle (p1,p2,p3) iff ( tricord1 (p1,p2,p3,p) < 0 or tricord2 (p1,p2,p3,p) < 0 or tricord3 (p1,p2,p3,p) < 0 ) )

proof end;

theorem Th56: :: EUCLID_3:56

for p1, p2, p3, p being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
( p in Triangle (p1,p2,p3) iff ( tricord1 (p1,p2,p3,p) >= 0 & tricord2 (p1,p2,p3,p) >= 0 & tricord3 (p1,p2,p3,p) >= 0 & ( tricord1 (p1,p2,p3,p) = 0 or tricord2 (p1,p2,p3,p) = 0 or tricord3 (p1,p2,p3,p) = 0 ) ) )

proof end;

theorem :: EUCLID_3:57

for p1, p2, p3, p being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
( p in Triangle (p1,p2,p3) iff ( ( tricord1 (p1,p2,p3,p) = 0 & tricord2 (p1,p2,p3,p) >= 0 & tricord3 (p1,p2,p3,p) >= 0 ) or ( tricord1 (p1,p2,p3,p) >= 0 & tricord2 (p1,p2,p3,p) = 0 & tricord3 (p1,p2,p3,p) >= 0 ) or ( tricord1 (p1,p2,p3,p) >= 0 & tricord2 (p1,p2,p3,p) >= 0 & tricord3 (p1,p2,p3,p) = 0 ) ) ) by Th56;

theorem Th58: :: EUCLID_3:58

for p1, p2, p3, p being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
( p in inside_of_triangle (p1,p2,p3) iff ( tricord1 (p1,p2,p3,p) > 0 & tricord2 (p1,p2,p3,p) > 0 & tricord3 (p1,p2,p3,p) > 0 ) )

proof end;

theorem :: EUCLID_3:59

for p1, p2, p3 being Point of (TOP-REAL 2) st p2 - p1,p3 - p1 are_lindependent2 holds
not inside_of_triangle (p1,p2,p3) is empty

proof end;