Rational right triangle tripleswith special linear relationship of areas and perimeters

Abstract

Suppose that the rational right triangle triple is $(T_1,T_2,T_3)$, their areas are $A_i~(i=1,2,3)$, and perimeters are $P_i~(i=1,2,3)$. By the theory of elliptic curves, we investigate the solvability of the following Diophantine system \[A_1+\alpha A_2=\beta A_3,\qquad P_1+\alpha P_2=\beta P_3,\]where $\alpha$ and $\beta$ are rational numbers. When $(\alpha,\beta)=(-2,-1)$ or $(\alpha,\beta)=(1,1)$, we show that there are infinitely many rational right triangle triples with the same perimeter and the areas in arithmetical progression or with the areas and perimeters satisfying the linear recurrence equation of Lucas sequence respectively. Moreover, we prove that there is no rational right triangle triple whose areas, perimeters and radii of the inscribed circles satisfy the linear recurrence equation of Lucas sequence respectively.