How to create Triple Phytagoras

Something that I like is math but can not be sad if I did not lecture in the majors. Last night when I was given the opportunity to provide math tutoring I find something new to me because previously I had never studied it, namely Triple Pytagoras.



As we know from Theorem Pytagoras expressed in a right triangle apply broad square on the hypotenuse (the side before the right angle of a right triangle) or the hypotenuse equals the total square area on two sides of a right angle clamp elbow.

As for the "Triple Pytagoras" is defined like this in a book I read: if a, b, c natural numbers is the length of the sides of a triangle, with hypotenuse c and satisfy the equation c² = a² + b², then the three numbers a, b, and c is called Triple Pytagoras and the triangle is a right triangle.


So there is a formula
If m > n maka m² + n², m² - n², 2mn are Triple Pytagoras

From there I got interested, using the formula I use Excel and be a table like this:

m	n	c	a	b
2	1	5	3	4
3	1	10	8	6
3	2	13	5	12
4	1	17	15	8
4	2	20	12	16
4	3	25	7	24
5	1	26	24	10
5	2	29	21	20
5	3	34	16	30
5	4	41	9	40
6	1	37	35	12
6	2	40	32	24
6	3	45	27	36
6	4	52	20	48
6	5	61	11	60
7	1	50	48	14
7	2	53	45	28
7	3	58	40	42
7	4	65	33	56
7	5	74	24	70
7	6	85	13	84
8	1	65	63	16
8	2	68	60	32
8	3	73	55	48
8	4	80	48	64
8	5	89	39	80
8	6	100	28	96
8	7	113	15	112
9	1	82	80	18
9	2	85	77	36
9	3	90	72	54
9	4	97	65	72
9	5	106	56	90
9	6	117	45	108
9	7	130	32	126
9	8	145	17	144
10	1	101	99	20
10	2	104	96	40
10	3	109	91	60
10	4	116	84	80
10	5	125	75	100
10	6	136	64	120
10	7	149	51	140
10	8	164	36	160
10	9	181	19	180

from there I was interested in studying some of which I consider similar triangles. With the reference values m and n terkumpullah several triangles the same as below:

2	1
3	1
4	2
6	2
8	4

or

3	2
5	1
6	4
10	2

or

4	1
5	3
8	2
10	6

and so on,
from there I can pull Conclusion I = triple pytagoras triangle will always form a triangle triple pytagoras comparable, with a size twice that of the triple pytagoras triangle at the previous level and the shape of reflection and rotation 90⁰ of the triple pytagoras triangle at the previous level.

After that I met also the uniqueness of the model, where the triangles are "equal" has its own unique formula. Thus formed is also  Conclusion II = if m and n are the authors of a triple pytagoras triangle, while a and b are the authors of a comparable pytagoras triple triangle on it, then
a = m + n,
dan
b = (m²-n²) : a
atau
b = √ a²-4mn 

And there have not finished my curiosity by forming the table below

m	n	c	a	b
2	1	5	3	4
3	2	13	5	12
4	1	17	15	8
4	3	25	7	24
5	2	29	21	20
5	4	41	9	40
6	1	37	35	12
6	3	45	27	36
6	5	61	11	60
7	2	53	45	28
7	4	65	33	56
7	6	85	13	84
8	1	65	63	16
8	3	73	55	48
8	5	89	39	80
8	7	113	15	112

From there I can make Conclusion III = To form a triple pytagoras triangle different with elements m and n where m> n then if m odd then n must be even or vice versa Juka m is even then n must be odd. 

Well that's the result of my adventure for 2 hours after knowing what it was Triple Pytagoras. May be useful for you (especially for math teachers to create questions pytagoras not in the form of fractions) ^_^