Vogeler's method for generating Pythagorean Triples
Pythagorean triples (a,b,c) are often calculated using the Euclid formula a=m²-n², b=2mn, and c=m²+n² where m and n are positive integers and n<m. If m and n are relatively prime and not both odd, the three numbers form a primitive Pythagorean triple (PPT). All PPT's can be generated (algebraically) from the (3,4,5) triple (m=2,n=1). The three PPT generated from (3,4,5) are (21,20,29), (5,12,13) and (15,8,7). The new PPT's are sometimes called the "children" of the original PPT. (See Wikipedia topic: Tree of Pythagorean triples.)
Vogeler's geometric method starts with the unit circle inside a square (see diagram). A line is drawn from the point P=(1,1) to W=(-1,0). The line intersects the unit circle in the point Q=(3/5,4/5). A rectangle QDFG is drawn inside the circle. P is now connected to D,F and G. These lines will intersect the unit circle in the 1st quadrant in points Q1,Q2 and Q3 to form the "children". See instruction below,
INSTRUCTION:Use the tick boxes to display the "children". Change the value of m and n (see new values generated), to generate new triples (a,b,c).
This construction is based on a diagram found in “The Book of Numbers, p. 172" written by John Conway and Richard Guy.