Integral Triangles with one angle twice another, and with the bisector(of the double angle) also of integral length

TL;DR

This paper characterizes integral triangles where one angle is twice another and the bisector of the double angle is also integral, providing new parametric formulas that describe the entire family of such triangles.

Contribution

It introduces new parametric formulas for all integral triangles with an angle twice another and with an integral bisector, expanding upon previous partial results.

Findings

Derived comprehensive parametric formulas for these triangles.

Extended previous work by providing a full family description.

Confirmed the conditions for the existence of such integral triangles.

Abstract

Let ABC be a triangle with a,b,and c being its three sidelengths. In a 1976 article by Wynne William Wilson in the Mathematical Gazette(see reference[2]), the author showed that angleB is twice angleA, if and only if b^2=a(a+c). We offer our own proof of this result in Proposition1.Using Proposition1 and Lemma2, we establish Proposition 2: Let a,b,c be positive reals. Then a triangle ABC having a,b,c as its sidelengths can be formed if,and onlyif, b^2=a(a+c) and either c<(or equal to)a; or alternatively a<c<3a. Now, consider the case of integral triangles, that is; a,b, and c bieng positive integers.In 2002, in a paper published in the Mathematical Gazette(see[2]), author M.N.Deshpande provided two-parameter formulas that describe some integral triangles with (angle)B=2(angle)A. In Result2 in Section5, we offer 3-parameter formulas that describe the entire family of integral triangles ABC with angleA=2angleB. Using Result1, we then parametrically describe the entire family of integral triangles with angle A=2angleB; and with the bisector of angleB also of integral length. This is done in Reult2 in Section6. In Section7, we conclude this article with two closing remarks.