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

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.