Infinite Integer Triangles
Here’s an interesting question.
Given the side of a triangle with integer length, what is the set of all points in the plane for which the other two sides will also have integer lengths?
And by interesting, I mean that the answer wasn’t immediately obvious to me.
So I drew a segment 5 units long in Geometer’s SketchPad, created a bunch of concentric circles with integer radii and centers at the endpoints of the segment, identified the intersection points of those circles, and finally hid the circles. The result was the following beautiful image:
And by beautiful, I mean that the result is, well, beautiful. At least to a math dork. If this had been painted by Van Gogh, it would have been called Triangle in a Starry Night. (Okay, maybe not.)
The triangle indicated by the dashed lines is the famous 3-4-5 right triangle. The points in the upper right and upper left corners yield the less well known but similarly intoxicating 5-5-8 triangle. If the limitations of the web allowed this image to extend infinitely in all directions, the result would be infinite beauty. Alas, reality confines us.
I have an infinity of jokes that deal with triangles and circles, but I’ll only share a subset of them here.
What did the triangle say to the circle?
Your life is pointless.
Why don’t circles hang out with ellipses?
What did the hypotenuse say to the other two sides?
Where do circles and ellipses spend their vacations?
What’s a circle?
A round, straight line with a hole in the middle.
What did the circle say to the tangent line?
Stop touching me!