Finding dense subgraphs in a graph is a fundamental graph mining task, with applications in several fields. Algorithms for identifying dense subgraphs are used in biology, in finance, in spam detection, etc. Standard formulations of this problem such as the problem of finding the maximum clique of a graph are hard to solve. However, some tractable formulations of the problem have also been proposed, focusing mainly on optimizing some density function, such as the degree density and the triangle density. However, maximization of degree density usually leads to large subgraphs with small density. In this paper, we introduce the k-clique-graph densest subgraph problem, k >= 3, a novel formulation for the discovery of dense subgraphs. Given an input graph, its k-clique-graph is a new graph created from the input graph where each vertex of the new graph corresponds to a k-clique of the input graph and two vertices are connected with an edge if they share a common (k-1)-clique. We define a simple density function, the k-clique-graph density, which gives compact and at the same time dense subgraphs, and we project its resulting subgraphs back to the input graph. In this paper we focus on the triangle-graph densest subgraph problem obtained for k=3. To optimize the proposed function, we present an efficient greedy approximation algorithm that scales well to larger graphs. We evaluate the proposed algorithm on real datasets and compare it with other algorithms in terms of the size and the density of the extracted subgraphs. The results verify the ability of the proposed algorithm in finding high-quality subgraphs in terms of size and density. Finally, we apply the proposed method to the important problem of keyword extraction from textual documents.