Incircles and Excircles of Pythagorean triangles
The post explores a geometric property of Pythagorean triangles: the radius of the incircle and the radii of the excircles are all integers. It connects this observation to earlier discussions on the Star Trek lemma and Pythagorean triples, highlighting the relationship between these geometric elements in right triangles with integer sides.
Background
- John D. Cook is a mathematician and blogger who writes about applied math, computer science, and number theory, often connecting seemingly unrelated topics.
- A "Pythagorean triangle" is a right triangle whose side lengths are a Pythagorean triple (e.g., 3-4-5), meaning all three sides are integers.
- The "incircle" of a triangle is the circle inscribed inside it, tangent to all three sides. An "excircle" is a circle tangent to one side and the extensions of the other two sides — each triangle has three excircles.
- A previous post ("Star Trek lemma") refers to a geometry result named jokingly after a scene in Star Trek: The Next Generation where Captain Picard recalls a geometric fact. The lemma states that in any triangle, the incircle radius r, the circumcircle radius R, and the distance d between the two centers satisfy d² = R(R – 2r).
- This post shows a neat number-theoretic property: for Pythagorean triangles, the incircle radius and the three excircle radii are all integers, and they relate in a simple way to the sides of the triangle.