II. O. Geometry Pythagorean Triples (Sierpinski);
Area = xy; more
Pythagorean Triples per Sierpinski page 8
and Proof of Right Triangle Area = xy
“Every primitive Pythagorean triple (a, b, c) where b is an even integer is
obtained only once per the following, where u and v are odd and u  v :
u 2  v2
u 2  v2
a  uv
b
c
2
2
Here, u and v represent all pairs of odd, relatively prime natural
numbers.” -- Waclaw Sierpinski, Pythagorean Triangles, The Scripta
Mathematics Studies Number Nine. New York City: Graduate School of
Science, Yeshiva University, 1962.
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Proof of Right Triangle Area = xy
Dr. Stan Hartzler
Archer City High School
x
x
y
r
r
r
Area =
y
1
1
r  perimeter = r (2r  2 x  2 y )  r (r  x  y )
2
2
( x  r )2  ( y  r )2  ( x  y)2
x 2  2 xr  r 2  y 2  2 yr  r 2  x 2  2 xy  y 2
2 xr  r 2  2 yr  r 2  2 xy
2 xr  2r 2  2 yr  2 xy
xr  r 2  yr  xy
r ( x  r  y )  xy  AREA
1
II. O. Geometry Pythagorean Triples (Sierpinski);
Area = xy; more
2
Fun Relationships
a  b  c v(u  v)
r


2
2
x 2  6 xy  y 2  ( x  y )
2
v 2  uv
2
2
u  uv
y br 
2
x  ar 
a  xr 
x 2  6 xy  y 2  x  y
2
b yr 
x 2  6 xy  y 2  x  y
2
c  x y
Perimeter = a  b  c  2(r  x  y )  x  y  x 2  6 xy  y 2  uv  u 2
Area =
ab
u 3 v  uv 3
 xy 
2
4
Developing u, v for a given r
Example: r = 3 × 5 × 7 = 105
r
uv  v 2
v 2  2r
v 2  210
u 
=
2
v
v
v
1
3
5
7
15
21
35
105
u
211
73
47
37
29
31
41
107
a
211
219
235
259
435
651
1435
11235
b
22260
2660
1092
660
308
260
228
212
c
22261
2669
1117
709
533
701
1453
11237
The number of distinct values of (u, v) for a given r of n odd prime factors
is
2n .
And note that
uk  vk  u2n 1k  v2n 1k .