A Diophantine $m$-tuple is a set of $m$ positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form $y^2=(ax+1)(bx+1)(cx+1)$, where $\{a,b,c\}$, is a Diophantine triple. In particular, we consider the elliptic curve $E_k$ defined by the equation $y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1),$ where $k\ge 2$ and $F_n$, denotes the $n$-th Fibonacci number. We prove that if the rank of $E_k\left(𝐐\right)$ is equal to one, or $k\le 50$, then all integer points on $E_k$ are given by$$\begin{array}{c}(x,y)\in \{(0\pm 1),(4F_{2k+1}{F}_{2k+2}{F}_{2k+3}\pm \left(2F_{2k+1}{F}_{2k+2}-1\right)\hfill \\ \hfill \times \left(2F_{2k+2}^2+1\right)\left(2F_{2k+2}{F}_{2k+3}+1\right)\}.\end{array}$$

We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.