We introduce a model which consists in a planar network which grows by adding nodes at a distance r from the pre-existing barycenter. Each new node position is randomly located through the distribution law P(r) oc 1/r γ with γ > 1. The new node j is linked to only one pre-existing node according to the probability law P(i ↔ j) ∝ η i k i /r αA ij (1 ≤ i < j; α A ≥ 0); k i is the number of links of the i th node, η i is its fitness (or quality factor), and Tij is the distance. We consider in the present paper two models for η i . In one of them, the single fitness model (SFM [D. J. B. Soares, C. Tsallis, A. M. Mariz and L. R. da Silva, Europhys. Lett. 70 (2005), 70.]), we consider η i = 1 Vi. In the other one, the uniformly distributed fitness model (UDFM), η i is chosen to be uniformly distributed within the interval (0,1]. We have determined numerically the degree distribution P(k). This distribution appears to be well fitted with P(k) = P(1) k -λ e -(k-1)/κ q with q ≥ 1, where e x q ≡ [1 + (1 - q)x] 1/(1-q) (e x 1 = e x ) is the q-exponential function naturally emerging within Tsallis nonextensive statistical mechanics. We determine, for both models, the entropic index q as a function of α A . Additionally, we determine the average topological (or chemical) distance within the network, and the time evolution of the average number of links (k i ). We obtain that, asymptotically, (k i ) ∝ (t/i) β , (i coincides with the input-time of the i th node) and β(α A ) to both cases.