46 Lessons in Early Geometry, part 2/10 / provisional version written in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch

Lesson 9

If a square measures 3 by 4 units, the diagonals measure 5 units. Let me prove this by means of a square whose side measures 3+4 or 4+3 = 7 units (please consider the points and capitals in the ASCII drawings below as dimensionless):

a a a . d d d d   i i i i I h h h

a a a . d d d d   i             h

a a a . d d d d   i             h

. . . . . . . .   F             h  oblique square FGHI

e e e . b b b b   f             H  side c, area cc

e e e . b b b b   f             g

e e e . b b b b   f             g

e e e . b b b b   f f f G g g g g

The two squares measure 7 by 7 units each:

a+b = 3+4 = b+a = 4+3 = 7   7x7 = 49

The left square consists of two small squares and rectangles:

aa = 3x3 = 9   bb = 4x4 = 16   ab = 3x4 = ba = 4x3 = 12

The right square consists of an oblique square (FGHI) and four rectangular triangles:

FGHI = cc = ??   a-b-c = 3-4-?   b-a-c = 4-3-?

The rectangles and the four triangles have the same area:

3x4 + 4x3 = 3x4/2 + 4x3/2 + 3x4/2 + 4x3/2 = 24

This means that also the remaining figures, namely the two small squares and the oblique square, have the same area:

aa + bb = cc    (49-24 =)  3x3 + 4x4 = 25 = 5x5

The side of the oblique square measures 5 units, exactly, with no mistake at all  proof 1

The above reasonings hold for any values of a and b  proof 2

aa + bb = cc   triple a-b-c  (rectangular triangle)

3x3 plus 4x4 equals 5x5   triple 3-4-5

6x6 plus 8x8 equals 10x10   triple 6-8-10

5x5 plus 12x12 equals 13x13  triple 5-12-13

8x8 plus 15x15 equals 17x17  triple 8-15-17

20x20 plus 21x21 equals 29x29  triple 20-21-29

The basic theorem of geometry can easily be expanded into 3 dimensions:

aa + bb + cc = dd   quadruple a-b-c-d

1x1 + 2x2 + 2x2 = 3x3    quadruple 1-2-2-3

8x8 + 9x9 + 12x12 = 17x17    quadruple 8-9-12-17

Lesson 10

My first number column for the approximate calculation of the square can also be used as a generator of triples, which again approximate the square:

1     1     2

2     3     4

5     7    10

12    17    24

29    41    58

70    99   140

169   239   338

408   577  ....

1   1   2      2x2  1x3  1x1+2x2

2   3   4     4    3      5

rectangular triangle 4-3-5,

periphery 3x4 = 12

area 1x1x2x3 = 6

radius of the inscribed circle 1x1 = 1

diameter 1x2 = 2

tangents of the half angles 1, 1/2, 1/3

rectangle 3x4

2   3   4       4x5  3x7  2x2+5x5

5   7   10     20   21     29

rectangular triangle 20-21-29

periphery 7x10 = 70

area 2x3x5x7 = 210

radius of the inscribed circle 2x3 = 6,

diameter 3x4 = 12

tangents of the half angles 1, 2/5, 3/7

rectangle 20x21

5    7    10       10x12  7x17  5x5+12x12

12   17    24     120    119     169

rectangular triangle 120-119-169

periphery 17x24 = 408

area 5x7x12x17 = 7140

radius of the inscribed circle 5x7 = 35

diameter 7x10 = 70

tangents of the half angles 1, 5/12, 7/17

rectangle 119x120

12    17    24       24x29  17x41  12x12+29x29

29    41    58     696    697       2545

rectangular triangle 696-697-2545

periphery 41x58 = 2378

area 12x17x29x41 = 242,556

radius of the inscribed circle 12x17 = 204

diameter 17x24 = 408

tangents of the half angles 1, 12/29, 17/41

rectangle 696x697

and so on

The resulting rectangles again approximate the square: 3x4 (diagonal 5), 20x21 (29), 119x120 (169), 696x697 (985), 4059x4060 (5741), 23660x23661 (33461) ...

Lesson 11

The number patterns for the calculation of the square can be started with any pair of numbers a and b, and every such number pattern can be used as a generator of triples:

a     b      2a

a+b   b+2a    2(a+b)

2     7     4

9    11    18

20    29    40

49    69    98

118   167   236

...   ...   ...

2   7    4       4x9  7x11  2x2+9x9

9   11   18     36    77       85

rectangular triangle 36-77-85

periphery 11x18 = 198

area 2x7x9x11 = 1386

radius of the inscribed circle 2x7 = 14

diameter 7x4 = 28

tangents of the half angles 1, 2/9, 7/11

9    11    18       18x20  11x29  9x9+20x20

20    29    40     360    319      481

rectangular triangle 360-319-481

periphery 29x40 = 1160

area 9x11x20x29 = 57420

radius of the inscribed circle 9x11 = 99

diameter 11x18 = 198

tangents of the half angles 1, 9/20, 11/29

and so on

1       999         2

1000      1001      2000

2001      3002      ....

2x1000, 999x1001, 1x1+2001x2001   triple 2000-999999-1000001

and so on

Lesson 12

The number columns for the calculation of the square also generate quadruples, integer solutions of right-parallelepipeds and their cubic diagonals:

1     1     2          2     7     4

2     3     4          9    11    18

5     7    10         20    29    40

12    17    ..         49    69    ..

1 1 2   1x2=2  1x1=1  1x2=2  1+2=3   quadruple 2-1-2-3

2 3 4   3x4=12  3x3=9  2x4=8  9+8=17   quadruple 12-9-8-17

5 7 10   7x10=70  7x7=49  5x10=50  49+50=99   q 70-49-50-99

and so on

2 7 4   7x4=28  7x7=49  2x4=8  49+8=57

9 11 18   11x18=198  11x11=121  9x18=162  121+162=283

20 29 40   29x40=1160  29x29=841  20x40=800  841+841=1641

and so on

In algebraic terms:

a   b    2a

a+b b+2a  2(a+b)

triple  2a(a+b) -- b(b+2a) -- aa+(a+b)(a+b)

=  2aa+2ab -- bb+2ab -- 2aa+2ab+bb

first quadruple  bx2a -- bb -- ax2a -- bb+(ax2a)

=  2ab -- bb -- 2aa -- bb+2aa

triple:    2aa + 2ab

2ab + bb

2aa + 2ab + bb

bb

2aa

2aa    +    bb

a=1, b=1, triple 4-3-5, quadruple 2-1-2-3

a=12, b=19, t 744-817-1105, q 288-456-361-649