000RM.dvi

6.4 Points at integer distances from the sides of a primitive Pythagorean triangle 213

6.4 Points at integer distances from the sides of a prim-

itive Pythagorean triangle

Let(a, b, c)be a primitive Pythagorean triangle, with vertices(a,0),
(0,b), and(0,0). The hypotenuse is the linebx+ay =ab. The dis-
tance of an interior point(x, y)to the hypotenuse is^1 c(ab−bx−ay).We
seek interior points which are at integer distances from the hypotenuse.
With the parameters (6,1) we have the Pythagorean triangle (35,12,37).
Here the five points (29,1), (23,2), (17,3), (11,4), (5,5) are at distances 1,
2, 3, 4, 5 from the hypotenuse.

12

0 35

4

3

2

1

5

Another example: with paramters (5,2) we have the triangle (21,20,29).
Here we have the interior points (8,11), (16,2), (3,13), (11,4), (6,6), (1,8),
(4,1) at distances 1, 2, 3, 4, 6, 8, 11 from the hypotenuse. The arrange-
ment is not as regular as the previous example.
20

0 21

8

3

11 4

1

2

6