The Seventeen Elements of Pythagorean Triangles

This is an exhaustive study of the seventeen elements of Pythagorean triangles, from the point of view of when such an element is an irrational number, a rational number, or an integer. For each of these 17 elements,precice conditions for their integrality,rationality, or irrationality; are given. These 17 elements are:The radius of the incircle, the radius of the circumscribed circle,the three radii of the three exterior circles tangential to the three lines containing the triangles’sides, the lengths of the three heights, the lenghts of the three internal angle bisectors, the lengthe of the three external angle bisectors, and the lengths of the three medians.

💡 Research Summary

The paper conducts a comprehensive investigation of seventeen geometric quantities associated with a right‑angled triangle, focusing on the precise conditions under which each quantity is an integer, a rational number, or an irrational number. The triangle is denoted by legs a and b (both taken as positive integers) and hypotenuse c = √(a² + b²). The study distinguishes two major regimes: (i) Pythagorean triples where c is an integer, and (ii) the generic case where c is irrational. For each of the seventeen elements—incircle radius r, circumradius R, the three ex‑radii rₐ, r_b, r_c, the three altitudes hₐ, h_b, h_c, the three internal angle bisectors ℓₐ, ℓ_b, ℓ_c, the three external angle bisectors, and the three medians mₐ, m_b, m_c—the author derives explicit algebraic formulas and then analyses the arithmetic nature of the resulting expressions.

Key formulas:

• Incircle radius: r = (a + b − c)/2.
• Circumradius: R = c/2.
• Ex‑radii: rₐ = (b + c − a)/2, r_b = (a + c − b)/2, r_c = (a + b + c)/2.
• Altitudes: h_c = ab/c (altitude to the hypotenuse), hₐ = b c/a, h_b = a c/b.
• Internal bisectors: ℓₐ = (2bc cos (A/2))/(b + c), with analogous expressions for ℓ_b, ℓ_c. In a right triangle cos (A/2) can be expressed as √