A 4-dimensional proof of Heron's Formula - University of South
Transcript of A 4-dimensional proof of Heron's Formula - University of South
![Page 1: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/1.jpg)
A 4-dimensional proof of Heron’s Formula
J. Scott CarterDavid Mullens
University of South Alabama
November 2010MAA Regional meeting, Pensacola, FL
![Page 2: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/2.jpg)
Purpose
Give 4-d proof of Heron’s formula
using ScissorsCongruences.
Thanks to organizers
Thanks to my undergraduate advisor J. Scott Carter
![Page 3: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/3.jpg)
Purpose
Give 4-d proof of Heron’s formula using ScissorsCongruences.
Thanks to organizers
Thanks to my undergraduate advisor J. Scott Carter
![Page 4: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/4.jpg)
Purpose
Give 4-d proof of Heron’s formula using ScissorsCongruences.
Thanks to organizers
Thanks to my undergraduate advisor J. Scott Carter
![Page 5: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/5.jpg)
Purpose
Give 4-d proof of Heron’s formula using ScissorsCongruences.
Thanks to organizers
Thanks to my undergraduate advisor J. Scott Carter
![Page 6: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/6.jpg)
Area of Polygons
Ayt = 12pr
( q , r )
( 0 , 0 ) ( p , 0 )
r
( p - q )
a b
c
![Page 7: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/7.jpg)
Area of Polygons
Ayt = 12pr, Ap = pr
( q , r )
( 0 , 0 ) ( p , 0 )
( p+q , r )
r
( p - q )
a b
c
![Page 8: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/8.jpg)
Area of Polygons
Ayt = 12pr, Ap = pr
( q , r )
( 0 , 0 ) ( p , 0 )
( p+q , r )
r
( p - q )
a b
c
![Page 9: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/9.jpg)
Area of Polygons
Ayt = 12pr, Ap = pr, Ar = pr
( q , r )
( 0 , 0 ) ( p , 0 )
( p , r )
r
( p - q )
a b
c
( 0 , r )
![Page 10: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/10.jpg)
Area of Polygons
Ayt = 12pr, Ap = pr, Ar = pr, Abt = 1
2pr
( q , r )
( 0 , 0 ) ( p , 0 )
( p , r )
r
( p - q )
a b
c
( 0 , r )
![Page 11: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/11.jpg)
Heron’s Formula
At =√s(s− a)(s− b)(s− c) where the semiperimeter,
s = (a + b + c)/2. We will keep a < b < c.
a b
c
Heron’s Formula can be written strictly in terms of its edgesBy squaring both sides and clearing the denominator, i.e.,f(a, b, c) = 16A2
t = (a+b+c)(−a+b+c)(a−b+c)(a+b−c)
![Page 12: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/12.jpg)
Heron’s Formula
At =√s(s− a)(s− b)(s− c) where the semiperimeter,
s = (a + b + c)/2. We will keep a < b < c.
a b
c
Heron’s Formula can be written strictly in terms of its edges
By squaring both sides and clearing the denominator, i.e.,f(a, b, c) = 16A2
t = (a+b+c)(−a+b+c)(a−b+c)(a+b−c)
![Page 13: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/13.jpg)
Heron’s Formula
At =√s(s− a)(s− b)(s− c) where the semiperimeter,
s = (a + b + c)/2. We will keep a < b < c.
a b
c
Heron’s Formula can be written strictly in terms of its edgesBy squaring both sides and clearing the denominator, i.e.,
f(a, b, c) = 16A2t = (a+b+c)(−a+b+c)(a−b+c)(a+b−c)
![Page 14: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/14.jpg)
Heron’s Formula
At =√s(s− a)(s− b)(s− c) where the semiperimeter,
s = (a + b + c)/2. We will keep a < b < c.
a b
c
Heron’s Formula can be written strictly in terms of its edgesBy squaring both sides and clearing the denominator, i.e.,f(a, b, c) = 16A2
t = (a+b+c)(−a+b+c)(a−b+c)(a+b−c)
![Page 15: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/15.jpg)
Scissors Congruence
Polygons A and A′ are scissors congruent if there existspolygon decomposition P1, P2, ..., Pn and Q1, Q2, ..., Qn of Aand A′ respectively such that Pi is congruent to Qj.
In short two polygons are scissors congruent if one can becut up and reassembled into the other. Let us denotescissors congruence by A ∼ sc A′
If we take the cross product of two polygons A and B in R2
then the resulting figure will exist in R4, i.e., A×B ∈ R4 isa 4-d hypersolid.
![Page 16: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/16.jpg)
Scissors Congruence
Polygons A and A′ are scissors congruent if there existspolygon decomposition P1, P2, ..., Pn and Q1, Q2, ..., Qn of Aand A′ respectively such that Pi is congruent to Qj.
In short two polygons are scissors congruent if one can becut up and reassembled into the other. Let us denotescissors congruence by A ∼ sc A′
If we take the cross product of two polygons A and B in R2
then the resulting figure will exist in R4, i.e., A×B ∈ R4 isa 4-d hypersolid.
![Page 17: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/17.jpg)
Scissors Congruence
Polygons A and A′ are scissors congruent if there existspolygon decomposition P1, P2, ..., Pn and Q1, Q2, ..., Qn of Aand A′ respectively such that Pi is congruent to Qj.
In short two polygons are scissors congruent if one can becut up and reassembled into the other. Let us denotescissors congruence by A ∼ sc A′
If we take the cross product of two polygons A and B in R2
then the resulting figure will exist in R4, i.e., A×B ∈ R4 isa 4-d hypersolid.
![Page 18: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/18.jpg)
Higher Dimensional Polytopes
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Higher Dimensional Polytopes
![Page 20: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/20.jpg)
Higher Dimensional Polytopes
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Scissors Congruence
Let A,A′, B, and B′ be polygons in R2, let A ∼ sc A′, andlet B ∼ sc B′.
Then ∃P1, ..., Pn, Q1, ..., Qm such that,
A = P1 ∪ · · · ∪ Pn and A′ =⋃
Pi
B = Q1 ∪ · · · ∪Qm and B′ =⋃
Qj
![Page 22: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/22.jpg)
Scissors Congruence
Let A,A′, B, and B′ be polygons in R2, let A ∼ sc A′, andlet B ∼ sc B′.
Then ∃P1, ..., Pn, Q1, ..., Qm such that,
A = P1 ∪ · · · ∪ Pn and A′ =⋃
Pi
B = Q1 ∪ · · · ∪Qm and B′ =⋃
Qj
![Page 23: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/23.jpg)
Scissors Congruence
Let A,A′, B, and B′ be polygons in R2, let A ∼ sc A′, andlet B ∼ sc B′.
Then ∃P1, ..., Pn, Q1, ..., Qm such that,
A = P1 ∪ · · · ∪ Pn and A′ =⋃
Pi
B = Q1 ∪ · · · ∪Qm and B′ =⋃
Qj
![Page 24: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/24.jpg)
Scissors Congruence
Let A,A′, B, and B′ be polygons in R2, let A ∼ sc A′, andlet B ∼ sc B′.
Then ∃P1, ..., Pn, Q1, ..., Qm such that,
A = P1 ∪ · · · ∪ Pn and A′ =⋃
Pi
B = Q1 ∪ · · · ∪Qm and B′ =⋃
Qj
![Page 25: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/25.jpg)
Scissors CongruenceLemma 1: A×B ∼ sc A′ ×B′
Lemma 2: The distributive law is a scissors congruence.
x
y
z
zx
zy
![Page 26: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/26.jpg)
Scissors CongruenceLemma 1: A×B ∼ sc A′ ×B′
Lemma 2: The distributive law is a scissors congruence.
x
y
z
zx
zy
![Page 27: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/27.jpg)
Scissors CongruenceLemma 1: A×B ∼ sc A′ ×B′
Lemma 2: The distributive law is a scissors congruence.
x
y
z
zx
zy
![Page 28: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/28.jpg)
Scissors CongruenceLemma 1: A×B ∼ sc A′ ×B′
Lemma 2: The distributive law is a scissors congruence.
.
.
.
.
.
.
.
.
.
.x
y
z
zx
zy y
z
zy
zy
zy
y
y
n
2
1
n
2
1
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Scissors CongruenceLemma 1: A×B ∼ sc A′ ×B′
Lemma 2: The distributive law is a scissors congruence.
.
.
.
.
.
.
.
.
.
.x
y
z
zx
zy
v
w
u
uv
uw
tv
tw
( t + u )( v + w )
y
z
zy
zy
zy
y
y
n
2
1
n
2
1
t
![Page 30: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/30.jpg)
Scissors CongruenceDistributive Law in R2
c
(a+b)c = ac + bc
a b
![Page 31: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/31.jpg)
Scissors CongruenceDistributive Law in R2, R3
c
(a+b)c = ac + bc
b
(a+b)c = ac + bc 22 2
a
a
b
c
![Page 32: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/32.jpg)
Scissors CongruenceDistributive Law in R2, R3, R4
c
(a+b)c = ac + bc
b
(a+b)c = ac + bc (a+b)c = ac + bc33 3
22 2
a
a
a
b b
c
c
![Page 33: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/33.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 34: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/34.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 35: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/35.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 36: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/36.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 37: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/37.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally
⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 38: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/38.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 39: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/39.jpg)
Scissors Congruence
Lemma 1: A×B ∼ sc A′ ×B′
Since A = P1 ∪ · · · ∪ Pn and B = Q1 ∪ · · · ∪Qm
then A×B = (P1 ∪ · · · ∪ Pn)× (Q1 ∪ · · · ∪Qm).
We also have that A′ =⋃Pi and B′ =
⋃Qj.
Hence,⋃m
j=1 Pi ×Qj = Pi ×B′ and finally⋃mj=1
⋃ni=1 Pi ×Qj = A′ ×B′
Ergo, A×B ∼ sc A′ ×B′.
![Page 40: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/40.jpg)
Scissors Congruence
Remember...
( q , r )
( 0 , 0 ) ( p , 0 )
( p , r )
r
( p - q )
a b
c
( 0 , r )( q , r )
( 0 , 0 ) ( p , 0 )
( p , r )
r
( p - q )
a b
c
( 0 , r )
![Page 41: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/41.jpg)
Scissors Congruence
Remember...
( q , r )
( 0 , 0 ) ( p , 0 )
( p+q , r )
r
( p - q )
a b
c
( q , r )
( 0 , 0 ) ( p , 0 )
( p+q , r )
r
( p - q )
a b
c
![Page 42: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/42.jpg)
Scissors CongruenceExample in R4 looks as follows.
Rectangle x Rectangle
![Page 43: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/43.jpg)
Scissors CongruenceExample in R4 looks as follows.
Rectangle x Rectangle
![Page 44: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/44.jpg)
Scissors CongruenceExample in R4 looks as follows.
Triangle x Triangle
![Page 45: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/45.jpg)
Scissors CongruenceExample in R4 looks as follows.
Trapezoid x Triangle
![Page 46: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/46.jpg)
Scissors CongruenceExample in R4 looks as follows.
Triangle x Trapezoid
![Page 47: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/47.jpg)
Scissors CongruenceExample in R4 looks as follows.
Trapezoid x Trapezoid
![Page 48: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/48.jpg)
Scissors CongruenceExample in R4 looks as follows.
Parallelogram x Parallelogram
![Page 49: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/49.jpg)
Scissors CongruenceExample in R4 looks as follows.
move tri x tri to ...
![Page 50: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/50.jpg)
Scissors CongruenceExample in R4 looks as follows.
... here.
![Page 51: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/51.jpg)
Scissors CongruenceExample in R4 looks as follows.
Move trap x tri...
![Page 52: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/52.jpg)
Scissors CongruenceExample in R4 looks as follows.
...here.
![Page 53: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/53.jpg)
Scissors CongruenceExample in R4 looks as follows.
trap x trap stays in place.
![Page 54: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/54.jpg)
Scissors CongruenceExample in R4 looks as follows.
Move tri x trap ...
![Page 55: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/55.jpg)
Scissors CongruenceExample in R4 looks as follows.
...here.
![Page 56: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/56.jpg)
Scissors CongruenceExample in R4 looks as follows.
![Page 57: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/57.jpg)
Scissors Congruence
A Scissors Congruence proof of the Pythagorean Theoremlooks as follows.
a2 + b2 = c2
a
a
b
b
![Page 58: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/58.jpg)
Scissors Congruence
A Scissors Congruence proof of the Pythagorean Theoremlooks as follows.
a2 + b2 = c2
~sca
a
b
ba
a
a
b-a
b
bc
c
b-a
![Page 59: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/59.jpg)
Scissors Congruence
A Scissors Congruence proof of the Pythagorean Theoremlooks as follows.
a2 + b2 = c2
~sc ~sca
a
b
ba
a
a
b-a
b
bc
c
b-a
c
c
c
c
ba
b-a
![Page 60: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/60.jpg)
Scissors Congruence
Why is this important?
Because each time we see the sum of squares,
z
z
x
y
z2 = x2 + y2
![Page 61: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/61.jpg)
Scissors Congruence
Why is this important?Because each time we see the sum of squares,
z
z
x
y
z2 = x2 + y2
![Page 62: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/62.jpg)
Scissors Congruence
Why is this important?Because each time we see the sum of squares,
z
z
v
vx
y
t
u
z2 = x2 + y2 and v2 = t2 + u2.
![Page 63: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/63.jpg)
Scissors Congruence
Now it is important to note,
z2v2 = (x2 + y2)(t2 + u2)
= x2(t2 + u2) + y2(t2 + u2)
= x2t2 + x2u2 + y2t2 + y2u2
is a scissor congruence by Lemma 2.
![Page 64: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/64.jpg)
Scissors Congruence
Now it is important to note,
z2v2 = (x2 + y2)(t2 + u2)
= x2(t2 + u2) + y2(t2 + u2)
= x2t2 + x2u2 + y2t2 + y2u2
is a scissor congruence by Lemma 2.
![Page 65: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/65.jpg)
Scissors Congruence
Now it is important to note,
z2v2 = (x2 + y2)(t2 + u2)
= x2(t2 + u2) + y2(t2 + u2)
= x2t2 + x2u2 + y2t2 + y2u2
is a scissor congruence by Lemma 2.
![Page 66: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/66.jpg)
Scissors Congruence
Now it is important to note,
z2v2 = (x2 + y2)(t2 + u2)
= x2(t2 + u2) + y2(t2 + u2)
= x2t2 + x2u2 + y2t2 + y2u2
is a scissor congruence by Lemma 2.
![Page 67: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/67.jpg)
Scissors Congruence
Now it is important to note,
z2v2 = (x2 + y2)(t2 + u2)
= x2(t2 + u2) + y2(t2 + u2)
= x2t2 + x2u2 + y2t2 + y2u2
is a scissor congruence by Lemma 2.
![Page 68: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/68.jpg)
Scissors Congruence
Once again recall...
( q , r )
( 0 , 0 ) ( p , 0 )
( p+q , r )
r
( p - q )
a b
c
which tells us that we can writea2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2.
![Page 69: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/69.jpg)
Back to Heron’s Formula
16A2t = (a + b + c)(−a + b + c)(a− b + c)(a + b− c)
= 2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
Since a2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2, everypiece of Heron’s Formula left over that contains an a and bcan be thought of as
z
z
v
vx
y
t
u
![Page 70: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/70.jpg)
Back to Heron’s Formula
16A2t = (a + b + c)(−a + b + c)(a− b + c)(a + b− c)
= 2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
Since a2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2, everypiece of Heron’s Formula left over that contains an a and bcan be thought of as
z
z
v
vx
y
t
u
![Page 71: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/71.jpg)
Back to Heron’s Formula
16A2t = (a + b + c)(−a + b + c)(a− b + c)(a + b− c)
= 2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
Since a2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2,
everypiece of Heron’s Formula left over that contains an a and bcan be thought of as
z
z
v
vx
y
t
u
![Page 72: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/72.jpg)
Back to Heron’s Formula
16A2t = (a + b + c)(−a + b + c)(a− b + c)(a + b− c)
= 2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
Since a2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2, everypiece of Heron’s Formula left over that contains an a and bcan be thought of as
z
z
v
vx
y
t
u
![Page 73: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/73.jpg)
Back to Heron’s Formula
16A2t = (a + b + c)(−a + b + c)(a− b + c)(a + b− c)
= 2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
Since a2 = (q2 + r2), b2 = ((p− q)2 + r2), and c2 = p2, everypiece of Heron’s Formula left over that contains an a and bcan be thought of as
z
z
v
vx
y
t
u
![Page 74: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/74.jpg)
Algebraic Proof
2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
= 2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]+2[(q2 + r2)(p2)
]−(q2 + r2)2 − ((p− q)2 + r2)2 − p4
=[2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]−((p− q)2 + r2)2
]+2[(q2 + r2)(p2)
]− ((q2 + r2)2 − p4
= 4r2p2 + (p− q)2[2(q2 + r2) + 2p2 − (p− q)2 − 2r2
]+2(q2 + r2)r2 − r4 + 2q2p2 − (q2 + r2)− p4
![Page 75: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/75.jpg)
Algebraic Proof
2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
= 2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]+2[(q2 + r2)(p2)
]−(q2 + r2)2 − ((p− q)2 + r2)2 − p4
=[2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]−((p− q)2 + r2)2
]+2[(q2 + r2)(p2)
]− ((q2 + r2)2 − p4
= 4r2p2 + (p− q)2[2(q2 + r2) + 2p2 − (p− q)2 − 2r2
]+2(q2 + r2)r2 − r4 + 2q2p2 − (q2 + r2)− p4
![Page 76: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/76.jpg)
Algebraic Proof
2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
= 2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]+2[(q2 + r2)(p2)
]−(q2 + r2)2 − ((p− q)2 + r2)2 − p4
=[2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]−((p− q)2 + r2)2
]+2[(q2 + r2)(p2)
]− ((q2 + r2)2 − p4
= 4r2p2 + (p− q)2[2(q2 + r2) + 2p2 − (p− q)2 − 2r2
]+2(q2 + r2)r2 − r4 + 2q2p2 − (q2 + r2)− p4
![Page 77: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/77.jpg)
Algebraic Proof
2a2b2 + 2a2c2 + 2b2c2 − a4 − b4 − c4
= 2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]+2[(q2 + r2)(p2)
]−(q2 + r2)2 − ((p− q)2 + r2)2 − p4
=[2[(q2 + r2)((p− q)2 + r2)
]+ 2[((p− q)2 + r2)(p2)
]−((p− q)2 + r2)2
]+2[(q2 + r2)(p2)
]− ((q2 + r2)2 − p4
= 4r2p2 + (p− q)2[2(q2 + r2) + 2p2 − (p− q)2 − 2r2
]+2(q2 + r2)r2 − r4 + 2q2p2 − (q2 + r2)− p4
![Page 78: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/78.jpg)
Algebraic Proof
= 4r2p2 + (p− q)2[q2 + 2pq + p2
]+ 2q2p2 − q4 − p4
= 4r2p2 + (p− q)2[(q + p)2
]−[q4 − 2q2p2 + p4
]= 4r2p2 +
[(p− q)2(q + p)2
]−[(p− q)2(q + p)2
]= 4r2p2
So now we can see that 16A2t = 4r2p2 ⇒ A2
t = 14r2p2
![Page 79: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/79.jpg)
Algebraic Proof
= 4r2p2 + (p− q)2[q2 + 2pq + p2
]+ 2q2p2 − q4 − p4
= 4r2p2 + (p− q)2[(q + p)2
]−[q4 − 2q2p2 + p4
]
= 4r2p2 +[(p− q)2(q + p)2
]−[(p− q)2(q + p)2
]= 4r2p2
So now we can see that 16A2t = 4r2p2 ⇒ A2
t = 14r2p2
![Page 80: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/80.jpg)
Algebraic Proof
= 4r2p2 + (p− q)2[q2 + 2pq + p2
]+ 2q2p2 − q4 − p4
= 4r2p2 + (p− q)2[(q + p)2
]−[q4 − 2q2p2 + p4
]= 4r2p2 +
[(p− q)2(q + p)2
]−[(p− q)2(q + p)2
]
= 4r2p2
So now we can see that 16A2t = 4r2p2 ⇒ A2
t = 14r2p2
![Page 81: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/81.jpg)
Algebraic Proof
= 4r2p2 + (p− q)2[q2 + 2pq + p2
]+ 2q2p2 − q4 − p4
= 4r2p2 + (p− q)2[(q + p)2
]−[q4 − 2q2p2 + p4
]= 4r2p2 +
[(p− q)2(q + p)2
]−[(p− q)2(q + p)2
]= 4r2p2
So now we can see that 16A2t = 4r2p2 ⇒ A2
t = 14r2p2
![Page 82: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/82.jpg)
Algebraic Proof
= 4r2p2 + (p− q)2[q2 + 2pq + p2
]+ 2q2p2 − q4 − p4
= 4r2p2 + (p− q)2[(q + p)2
]−[q4 − 2q2p2 + p4
]= 4r2p2 +
[(p− q)2(q + p)2
]−[(p− q)2(q + p)2
]= 4r2p2
So now we can see that 16A2t = 4r2p2 ⇒ A2
t = 14r2p2
![Page 83: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/83.jpg)
A2t =
14r
2p2
In fact Heron’s Formula tells us....
![Page 84: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/84.jpg)
A2t =
14r
2p2
In fact Heron’s Formula tells us....
x ≤ y w ≤ z
![Page 85: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/85.jpg)
A2t =
14r
2p2
In fact Heron’s Formula tells us....
( 0 , 0 ) ( 0 , 0 )
x ≤ y z ≤ w
x ≤ y w ≤ z
![Page 86: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/86.jpg)
A2t =
14r
2p2
In fact Heron’s Formula tells us....
( 0 , 0 ) ( 0 , 0 )
x ≤ y z ≤ w
y ≤ x w ≤ z
x ≤ y w ≤ z
![Page 87: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/87.jpg)
A2t =
14r
2p2
In fact Heron’s Formula tells us....
( 0 , 0 ) ( 0 , 0 )
x ≤ y z ≤ w
y ≤ x w ≤ z
x ≤ y w ≤ z
y ≤ x z ≤ w
![Page 88: A 4-dimensional proof of Heron's Formula - University of South](https://reader031.fdocuments.in/reader031/viewer/2022020706/61fc8af98d33c02b785e64e7/html5/thumbnails/88.jpg)