A Graphic Approach to Pythagorean Triangles using the Unit Circle

  

I wrote a couple of posts a while back on the Barning Tree method of finding Pythagorean triples using matrices, and then a followup(now included in previous post). I recently came across another approach to Pythagorean triples that involves a clever relationship between points on the positive y-axis, and points on the unit circle. (ok, maybe I should have known this, but I didn't.. and I think it is really a neat idea)

On the unit circle, x² + y ² = 1, if we draw a secant from the point (-1,0) through (0,t) on the y-axis, it turns out that if t is a rational number, then the coordinates of P=(x,y) where the secant intersects the circle, will also be rational. Since the slope of the line is also t, the equation is y=tx+t ... and so and t²= (1-x²)/(1+x)². 

That means t = y/(x+1) which leads to x= (1-t^2)/(1+t^2) and y= ^(2t)/_(1+t^2) If we pick some rational number to be t, say t=2/7,

then x= ⁴⁵/₅₃ and y= ²⁸/₅₃.... Then by similar triangles, there must be a circle with radius 53 and a point on the circle would be (28, 45) and in fact 28²+45²=53²... and any such rational point will produce another.